56-App-Answers.Rmd

# Answers to odd-numbered exercises {#Answers .unnumbered .unlisted}

```{r evenAnswers, message=FALSE, warning=FALSE, include=TRUE}
includeEvenAnswers <- FALSE
includeGraphsTables <- FALSE
```

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<!-- This Appendix contains answers to *most* (not all) exercises. -->
<!-- Some are fully worked, and some are only brief solutions. -->

<!-- * Answers to Chap.\ \@ref(Intro) (Introduction): Sect.\ \@ref(IntroAnswer). -->
<!-- * Answers to Chap.\ \@ref(RQs) (Research questions): Sect.\ \@ref(RQsAnswer). -->
<!-- * Answers to Chap.\ \@ref(ResearchDesign) (Research design): Sect.\ \@ref(ResearchDesignAnswer). -->
<!-- * Answers to Chap.\ \@ref(Ethics) (Ethics): Sect.\ \@ref(EthicsAnswer). -->
<!-- * Answers to Chap.\ \@ref(Sampling) (Sampling): Sect.\ \@ref(SamplingAnswer). -->
<!-- * Answers to Chap.\ \@ref(ResearchDesignOverview) (Factors that influence the response variable): Sect.\ \@ref(ResearchDesignOverviewAnswer). -->
<!-- * Answers to Chap.\ \@ref(DesignExperiment) (Designing experiments): Sect.\ \@ref(DesigningExperimentsAnswer). -->
<!-- * Answers to Chap.\ \@ref(DesignObservational) (Designing observational studies): Sect.\ \@ref(DesigningObservationalAnswer). -->
<!-- * Answers to Chap.\ \@ref(Interpretation) (Interpretation): Sect.\ \@ref(InterpretationAnswer). -->
<!-- * Answers to Chap.\ \@ref(CollectingDataProcedures) (Collecting data): Sect.\ \@ref(CollectionAnswer). -->
<!-- * Answers to Chap.\ \@ref(DescribingVars) (Describing variables): Sect.\ \@ref(DescribeAnswer). -->
<!-- * Answers to Chap.\ \@ref(Graphs) (Graphs): Sect.\ \@ref(GraphsAnswer). -->
<!-- * Answers to Chap.\ \@ref(NumericalQuant) (Numerical summaries for quantitative data): Sect.\ \@ref(NumericalQuantAnswer). -->
<!-- * Answers to Chap.\ \@ref(NumericalQual) (Numerical summaries for qualitative data): Sect.\ \@ref(NumericalQualAnswer). -->
<!-- * Answers to Chap.\ \@ref(MakingDecisions) (Making decisions): Sect.\ \@ref(MakingDecisionsAnswer) -->
<!-- * Answers to Chap.\ \@ref(Probability) (Probability): Sect.\ \@ref(ProbabilityAnswer). -->
<!-- * Answers to Chap.\ \@ref(SamplingDistributions) (Sampling distributions): Sect.\ \@ref(SamplingDistributionsAnswer). -->
<!-- * Answers to Chap.\ \@ref(SamplingVariation) (Sampling variation): Sect.\ \@ref(SamplingVariationExercisesAnswer). -->
<!-- * Answers to Chap.\ \@ref(CIOneProportion) (CIs for one proportion): Sect.\ \@ref(CIOneProportionAnswer). -->
<!-- * Answers to Chap.\ \@ref(AboutCIs) (More about forming CIs): Sect.\ \@ref(AboutCIsAnswer). -->
<!-- * Answers to Chap.\ \@ref(OneMeanConfInterval) (CIs for one mean): Sect.\ \@ref(OneMeanConfIntervalAnswer). -->
<!-- * Answers to Chap.\ \@ref(PairedCI) (CIs for paired data): Sect.\ \@ref(PairedCIExercisesAnswer). -->
<!-- * Answers to Chap.\ \@ref(CITwoMeans) (CIs for two independent means): Sect.\ \@ref(MeansIndSamplesAnswer). -->
<!-- * Answers to Chap.\ \@ref(OddsRatiosCI)  (CIs for odds ratios): Sect.\ \@ref(OddsRatiosCIAnswer). -->
<!-- * Answers to Chap.\ \@ref(EstimatingSampleSize) (Sample size estimation): Sect.\ \@ref(EstimatingSampleSizeAnswer) -->
<!-- * Answers to Chap.\ \@ref(TestOneProportion) (Tests for one proportion): Sect.\ \@ref(TestOneProportionAnswer). -->
<!-- * Answers to Chap.\ \@ref(TestOneMean) (Tests for one mean): Sect.\ \@ref(TestOneMeanAnswer). -->
<!-- * Answers to Chap.\ \@ref(MoreAboutTests) (More about hypothesis tests): Sect.\ \@ref(MoreAboutTestsAnswer). -->
<!-- * Answers to Chap.\ \@ref(TestPairedMeans) (Tests for paired mean): Sect.\ \@ref(TestPairedMeansAnswer). -->
<!-- * Answers to Chap.\ \@ref(TestTwoMeans) (Tests for two independent mean): Sect.\ \@ref(TestTwoMeansAnswer). -->
<!-- * Answers to Chap.\ \@ref(TestsOddsRatio) (Tests for odds ratios): Sect.\ \@ref(TestsOddsRatioAnswer). -->
<!-- * Answers to Chap.\ \@ref(TwoQuant) (Relationships between two quantitative variables): Sect.\ \@ref(TwoQuantAnswer). -->
<!-- * Answers to Chap.\ \@ref(Correlation)  (Correlation): Sect.\ \@ref(CorrelationAnswer). -->
<!-- * Answers to Chap.\ \@ref(Regression) (Regression): Sect.\ \@ref(RegressionAnswer). -->
<!-- * Answers to Chap.\ \@ref(Reading) (Reading research): Sect.\ \@ref(ReadAnswer). -->
<!-- * Answers to Chap.\ \@ref(WritingResearch) (Writing research): Sect.\ \@ref(WriteAnswer). -->


### Chap.\ \@ref(Intro): Research an introduction {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RQsTypeTourniquet).**
**1.** From many people: type of tourniquet; time to apply. 
**2.** **Quant**itative.
:::


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**Ex.\ \@ref(exr:RQsTypeMangroves).**
**1.** For many people: reason(s).
**2.** **Qual**itative.
:::
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RQsTypeSideEffects).**
**1.** For many Egyptians: whether side effects experienced after medication.
**2.** **Quant**itative.
:::


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**Ex.\ \@ref(exr:RQsTypeSolarPanels).**
**1.** For many homes: number of solar panels.
**2.** **Quant**itative.
:::
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RQsTypeGreenery).**
**Qual**itative
:::


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**Ex.\ \@ref(exr:RQsTypeGreenHydrogen).**
Mixed methods: interview and workshops qualitative; survey: quantitative.
:::
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::::::





### Chap.\ \@ref(RQs): Research questions {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RQsOutcomeResponse1).**
**1.** *Percentage* of vehicles that crash.
**2.** *Average* jump height.
**3.** *Average* number of tomatoes per plant.
:::


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**Ex.\ \@ref(exr:RQsOutcomeResponse2).**
**1.** *Percentage* people who own a car.
**2.** *Average* time for seedlings to sprout.
**3.** *Average* amount of caffeine in soft drinks.
:::
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RQsComparisonExplanatory1).**
**1.** Whether diet is vegan.
**2.** Whether coffee is caffeinated
**3.** Num. iron tablets/day.
:::


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**Ex.\ \@ref(exr:RQsComparisonExplanatory2).**
**1.** Whether the vegetables frozen or fresh.
**2.** Type of fuel.
**3.** Size of city.
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RQsComparisonVsPaired1).**
**1.** *Between*-individuals.
       Outcome: percentage wearing hats.
**2.** *Between*-intervals.
       Outcome: average yield (in kg/plant, tomatoes/plant, etc.).
:::

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**Ex.\ \@ref(exr:RQsComparisonVsPaired2).**
**1.** *Within*-individuals.
       Outcome: average balance times.
**2.** *Within*-individuals comparison.
       Outcome: average cholesterol levels.
:::
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RQsDogs).**
**1.** Correlational.
**2.** Neither variable is explanatory, response.
:::

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**Ex.\ \@ref(exr:RQsTypingCor).**
**1.** Correlational.
**2.** Age probably impacts typing speed; age: explanatory; typing speed: response.
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RQsBloodPressure).**
**1.** P: Danish Uni students; O: *average* resting diastolic blood pressure; C: between students who regularly drive, ride their bicycles to uni.
**2.** No intervention.
**3.** Relational.
**4.** Decision-making.
**5.** *Conceptual*: 'regularly'; 'university student' (on-campus? undergraduate?). *Operational*: how 'resting diastolic blood pressure' measured.
**6.** Resting diastolic blood pressure; whether they regularly drive, ride to uni.
**7.** Both:  Danish uni students.
:::


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**Ex.\ \@ref(exr:RQsNutrition).**
**1.** Some elements not well defined; perhaps: P: Children aged under\ $3$ in a Peruvian peri-urban community; O: proportion of children with diarrhoea; C: nutritional status; no intervention.
**2.** Perhaps: 'In children aged under\ $3$ in a Peruvian peri-urban community, is there a relationship between diarrhoea status and nutritional status?'.
**3.** 'Nutritional status' not well defined:amount intake per day? Then, correlational probably.
**4.** Decision-making.
**5.** 'Diarrhoea status'; 'nutritional status'; probably others.
**6.** Response: diarrhoea status; explanatory: nutritional status.
**7.** Both: the children.
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**Ex.\ \@ref(exr:RQsWalkingSpeed).**
**1.** Probably relational.
**2.** Two-tailed.
**3.** Probably not, but possible.
**4.** How *individual* people using phones ('Talking'; 'texting').
**5.** Walking speed.
**6.** *Average* walking speed.
**7.** Observation: person.
       Analysis: individuals not in a group, as people in group may not walk independently of  others (i.e., they keep pace with each other). 
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**Ex.\ \@ref(exr:RQsCommonCold).**
**1.** P: Japanese adults. 
       O: Between those who take, do not take Vit.\ C tablets.
       C: *Average* cold duration'.
**2.** Duration cold symptoms for each person.
**3.** Whether each person takes Vit.\ C tablets.
**4.** Decision-making
**5.** One-tailed.
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**Ex.\ \@ref(exr:RQsAnimals).**
**1.** Animal.
**2.** *Pen*: food allocated to pen.
Animals in same pen *not* independent: compete for same space, food, resources, have similar environments.
**3.** Between diets.
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**Ex.\ \@ref(exr:RQsBlueGum).**
False.
True.
True.
Sample size: $20$.
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<!-- SEM 2 2018: Christopher WALTON; WILLIUAM LUDOWYK; LOK RAJ BHATTA--> 
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ProjectRQ2).**
Ten adults is sample.
Unclear how many (or which) fonts compared.
Perhaps: 'Among Australian adults, does average time taken to read passage of text differ when Arial font used compared to Times Roman font?'
:::


<!-- SEM 2 2018: Alex Bruce; Levi Crawley; Stuart Stevens-->
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**Ex.\ \@ref(exr:ProjectRQ3).**
Outcome: *average* lung capacity.
Perhaps: 'For students that study at (University), do M have larger *average* lung capacity than F?'
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RQsNoseHair).**
**1.** Analysis: person; Observation: individual nose hairs.
Each unit of analysis has $50$ units of observation.
**2.** $n = 2$.
:::

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**Ex.\ \@ref(exr:RQsenvironments).**
**1.** Between environments.
**2.** *Before* and *after* for each individual.
**3.** Operational.
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**Ex.\ \@ref(exr:POCIaccelerometer).**
**1.** P: American adults; individuals: American adults.
**2.** O: average number recorded steps.
**3.** Response: number steps recorded for individuals.
       Explanatory: location of accelerometer.
**4.** *Within* individuals.
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**Ex.\ \@ref(exr:RQsTypesZinc).**
**1.** Descriptive; estimation.
**2.** Descriptive; decision-making.
**3.** Correlational; estimation.
**4.** Relational; decision-making.
      Intervention unlikely.
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**Ex.\ \@ref(exr:RQsComparisonConnectionCaloric).**
**1.** Relational; decision-making.
**2.** Correlational; estimation.
       Intervention unlikely.
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**Ex.\ \@ref(exr:InterventionalRQWithin).**
**1.** Relational; estimation.
**2.** Relational; decision-making.
Intervention (whether worms used).
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**Ex.\ \@ref(exr:WhatAreUnitsAnalysisA).**
$n = 27$; unit of analysis: emu.
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**Ex.\ \@ref(exr:WhatAreUnitsAnalysisB).**
$n = 155$; unit of analysis: person.
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:UnitsAnalysis).**
Unit of observation: tyre.
Unit of analysis: car (brand allocated to car; each car gets one brand).
Tyres on any car exposed to same day-to-day use, drivers, distances, etc.
Each unit of analysis produces four units of observations.
*Sample size*: $10$ cars ($40$ observations).
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**Ex.\ \@ref(exr:UnitsAnalysis2).**
Analysis: $12$ subjects.
Observation: $6$ per subjects: $72$.
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:UnitsAnalysisBamboo).**
**1.** Board.
**2.** $5$.
**3.** $10$.
**4.** $10$..
**5.** Within-board variation smaller (except first).
**6.** $1$.
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**Ex.\ \@ref(exr:PoorRQs).**
**1.** Should be worded as averages.
**2.** Everyone dies; time-frame needed (i.e., within $12$\ months of amputation). 
Compared to what?
**3.** No population (frozen beans? fresh butter beans? tinned kidney beans? coffee beans? can sizes?); "amount" of salt should be (e.g.) "average" amount (or concentration) of salt.
Anyway: read label (unless you wish to query those values).
**4.** Silly: elephants huge, joeys tiny; test not needed. 
Also, RQ talks has 'weight', not 'mean weight'.
**5.** More specific! Perhaps: 'Is the mean reaction time different for M and\ F?'.
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### Chap.\ \@ref(ResearchDesignOverview): Overview of research design {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MineArsenic).**
**1.** Arsenic conc.
**2.** Distance of lake from mine.
**3.** No: recorded.
**4.** Yes: may be related to response, explanatory variables.
**5.** Probably confounding.
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**Ex.\ \@ref(exr:ResearchDesignOverviewStudy1).**
*Response*: depression or not.
*Explanatory*: diet quality.
*Extraneous*: presumably *all* listed; that is why the researchers obtained this information.
*None* can be lurking variables: researchers measure or observe them.
*Confounding* variable potentially related to *both* the response *and* explanatory variables.
Hence, all of the extraneous variables could potentially be confounding variables.
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchDesignOverviewStudy2).**
*Response*: perhaps 'whether woman develops cancer of digestive system'.
*Explanatory*: 'whether participants drank green tea at least $3$\ times per week'.
*Lurking*: 'health consciousness of participants' (appears unrecorded).
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**Ex.\ \@ref(exr:ResearchDesignOverviewStudy3).**
Older children more likely to be smokers, would be larger in general: age a confounder.
Age easy to record, usually *is* recorded: probably *not* a lurking variable.
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**Ex.\ \@ref(exr:ResearchDesignOverviewRTEC).**
**1.** Sex; number siblings.
**2.** No.
**3.** Yes.
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**Ex.\ \@ref(exr:DesignExpExtraneous).**
Lurking; confounding.
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**Ex.\ \@ref(exr:DesignFindConfounderA).**
Age of person.
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**Ex.\ \@ref(exr:DesignFindConfounderB).**
Farmers with orchards having few insect problems with their current fertiliser not likely to buy new, more expensive, insecticide.
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**Ex.\ \@ref(exr:DesignConfoundingVarA).**
**1.** Soil quality; climate; size.
Control var.: perhaps farm size (e.g., farms over certain size).
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**Ex.\ \@ref(exr:DesignConfoundingVarB).**
**1.** Confounder.
**2.** Related perhaps to high BP.
**3.** Irrelevant. 
**4.** Perhaps confounder.
**5.** Non-confounder extraneous.
**6.** Response variable.
**7.** Maybe confounder.

Control: perhaps age.
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\pagebreak <!--               MANUAL           -->




### Chap.\ \@ref(ResearchDesign): Types of research studies {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TypesOfDesignsAcuteOtitis).**
**1.** Between-individuals.
**2.** Relational.
**3.** Most likely.
**4.** Estimation.
**5.** Intervention: experiment.
       Likely true experiment.
:::


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**Ex.\ \@ref(exr:TypesOfDesignsWorms).**
**1.** Between-individuals.
**2.** Relational, if C between individuals.
**3.** Yes.
**4.** Decision-making.
**5.** Experimental. True experiment.
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**Ex.\ \@ref(exr:ResearchDesignConcreteBeams).**
True experiment.
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**Ex.\ \@ref(exr:ResearchDesignVAP).**
**1.** Relational.
**2.** Between-individuals if relational.
**3.** Yes: program imposed.
**4.** Experimental.
       Program given to paramedics in certain cities; quasi-experimental (researchers cannot tell paramedics where to work).
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**Ex.\ \@ref(exr:ResearchDesignMatresses).**
Quasi-experiment.
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**Ex.\ \@ref(exr:ResearchDesignDietsForWeightLoss).**
**1.** Diet.
**2.** Change in body weight after $2$\ ys.
**3.** Experimental: diets *manipulated*, *imposed* by researchers.
**4.** Probably true experiment.
**5.** Individuals: diets allocated to individuals.
**6.** Individuals: those from whom the weight change is taken.
**7.** *Change* in body weight over $2$\ ys.
**8.** Type of diet.
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchDesignPetsAndHealth).**
**1.** Answers vary.
**2.** Researchers *intervene*: *give*, *not give* subjects pet.
**3.** Researchers *do not intervene*: subjects already *do*, *do not* own pet.
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::: {.exercise #ResearchDesignSeedsSprout}
**1.** Answers vary.
**2.** Researchers *intervene*: researchers *use*, *do not use* seed-raising mix.
**3.** Researchers *do not intervene*: find people *using*, *not using* seed-raising mix.
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::::::


### Chap.\ \@ref(Ethics): Ethics in research {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:EthicsCougars).**
Answers vary.
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**Ex.\ \@ref(exr:EthicsLying).**
Answers vary.
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**Ex.\ \@ref(exr:EthicsSideEffects).**
Answers vary.
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**Ex.\ \@ref(exr:EthicsLying).**
Answers vary.
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### Chap.\ \@ref(Sampling): External validity: sampling {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingAdvantageRandom).**
c. Externally-valid study more likely.
:::



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**Ex.\ \@ref(exr:SamplingAdvantageLarge).**
d. Precise estimates more likely.
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**Ex.\ \@ref(exr:SamplingOverUnderA).**
**1.** Under: practicality; intentional.
**2.** Under; deception; probably intentional.
**3.** Over; deception; intentional (cherry-picking).
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**Ex.\ \@ref(exr:SamplingOverUnderB).**
**1.** Under; deception; probably intentional.
**2.** Probably under; accidental; probably unintentional.
**3.** Under; deception; intentional.
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**Ex.\ \@ref(exr:SamplingSystematicProblem).**
**1.** Every $7$th day is same day of week.
**2.** Maybe select days at random over three-months.
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**Ex.\ \@ref(exr:SamplingBooks).**
Remember: some books borrowed (not physically in library).
**1.** Simple random sample: ist of all books held by library needed.
*May* be possible; probably not for non-library staff member.
In principle: number each book, randomly select sample from that list.
**2.** Stratified: campuses as strata; random sample of all books in each location.
**3.** Cluster: each shelf as cluster; randomly select some shelves; determine number of pages in each book on selected shelves.
**4.** Convenience: finding easily accessible books in libraries. 
**5.** Multi-stage: random sample of campuses; random sample of shelves in selected libraries; random shelf from each; a small number of random books from each shelf.
**6.** Multi-stage perhaps?
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**Ex.\ \@ref(exr:SamplingApartments).**
**1.** Multi-stage.
**2.** Stratified (floor), then convenience.
**3.** Convenience.
**4.** Part stratified (floors), then convenience.
Use first.
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**Ex.\ \@ref(exr:SamplingShoppingCentre).**
**1.** Convenience; approach every $10$th person may be little more representative.
**2.** Convenience; approach every $5$th person, go *every day* for a week, trying to make little more representative (and better than the first)
**3.** Self-selecting.
**4.** Convenience: By going every day for $2$\ weeks, at different times, locations each week, approaching someone every $15\mins$, to make more representative.

The fourth is best, still far from 'random'.
None truly random.
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**Ex.\ \@ref(exr:SamplingSchools).**
Random sampling (schools), then, self-selecting.
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**Ex.\ \@ref(exr:SamplingMalaria).**
True; False; True.
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingForest).**
Stratified by zone; then convenience.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:BiasOnline).**
No answer (yet).
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NotMultistageSampling).**
Stage\ $3$ selection *not* random.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NotClusterSampling).**
Stage\ $1$ selection of schools *not* random.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SolarSampling).**
**1.** Households in Santiago.
**2.** (c): all households in Santiago.
**3.** Voluntary response.
**4.** Multi-stage.
:::

::::::




### Chap.\ \@ref(DesignInternal): Internal validity {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DesignTrueFalse).**
All false.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DesignTrueFalse2).**
False; true; false; false; true.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DesignExpImproveIV).**
All but random samples (*external*).
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DesignExpImproveIV).**
All but random samples (*external*).
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchDesignHawthorneEffect).**
Also possible in observational studies.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:HawthorneTeeth).**
Yes; all subjects probably brushed better than usual.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchDesignObsPollen).**
In case hive size a confounder.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.exercise #ResearchDesignSeptic}
The control, acting as base.
Ethical.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DesignExpWeightLoss).**
Statements\ 1, 3, 4 and\ 8 true.
'Sex', 'Initial weight' *possible* confounders.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchSmokingAlfresco).**
**1.**. Observational.
**2.** PM~$2.5$~ concentration; number smokers.
**3.** Possibly: wind speed; amount cover.
**4.** Yes: be discreet.
**5.** No.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchDesignSunscreen).**
**1.** Observational.
**2.** *Response*: amount sunscreen used; *explanatory*: time applying sunscreen.
**3.** Potential extraneous, confounding variables.
**4.** To determine if sex a *confounder*.
**5.** Participants, researchers blinded.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DesignExpParamedicsPills).**
**1.** Experimental.
**2.** A group receiving pill like Treatment\ A and\ B, with no effective ingredient.
**3.** Blinding participants..
**4.** To ensure participants do not change behaviour because of the treatment.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchIcelandicGood).**
Random allocation; exclusion criteria; blinding; comparing two groups; ethical.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchBodyCameras).**
Blocking (matching); random allocation; comparing two groups.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchTanzaniaFarm).**
**1.** A: no blinding; B: double-blind.
**2.** Hawthorne effect impacting internal validity.
**3.** B; no Hawthorne effect.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchAustriaPA).**
**1.** Subjective: accelerometer.
**2.** Perceptions unreliable; Hawthorne effect.
**3.** Using accelerometer.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchTasteOfWater2).**
Randomly allocate type of water to subjects (or the *order* that subjects taste *each* drink.)
Subjects blind to water type.
Person providing water and receiving ratings blind to water type.
Random or representative sampling is good (but hard).
Use third-party if possible.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchDesignTasteOfWater).**
Random allocation not possible (observational).
Probably infeasible to blind participants, blind researchers.
Random or representative sample good, but hard. 
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchDesignFertilizer).**
Carryover effect; observer effect.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchDesignDust).**
**1.** Random allocation; blocking; recording potential confounders.
**2.** Blinding participants, researchers.
**3.** Change in nasal congestion.
**4.** Type of cleaning.
**5.** Age; sex.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ResearchDesignFormwork).**
**1.** Floor area.
**2.** Hours labour.
**3.** Extraneous.
**4.** Analysis
**5.** F.
**6.** T.
**7.** F.
**8.** F.
:::

::::::




### Chap.\ \@ref(Interpretation): Research design limitations {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ValidityLighting).**
External.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:InterpretationExerciseValidities).**
Ecological; external; internal.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:InterpretationExerciseExternalValidity).**
Population is students; external validity if applies to students at UniX not wider.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:InterpretationExerciseParachutes).**
Not *ecologically valid* (e.g., parachutes used at high altitude).
Probably not *externally valid*: voluntary (presumably convenience) sample.
*Internal validity* may be fine.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:InterpretationSleep).**
Sample not random; researchers (rightly) state results may not *generalise* to all hospitals.
Data collected at night; not *ecologically valid*?
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:LimitationsShopping).**
Ecological poor.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:InterpretationCoughDrops).**
Observational study: people with severe cough may take more cough drops.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:StatementsWithErrors).**

**1.** Observational: researchers cannot determine *comparison* (not outcome).
**2.** Mix of both C ('smokers and non-smokers') and O ('median serum cholesterol').
**3.** External validity refers to whether sample represents target population (Australians). Whether results apply for entire world is irrelevant.
**4.** 'Serum cholesterol' not a variable; nothing varies. 
Variable is 'serum cholesterol concentration'.
**5.** Not an experiment.
**6.** Data file needs $2$\ columns (variables): one records smoking status (Yes/No), one records serum cholesterol concentration.
**7.** Confounder must be related to *both* response and explanatory variables.
**8.** Observer effect about how researchers might respond, not individuals.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ValidityBeer).**
Lacks *ecological validity*.
:::

::::::




### Chap.\ \@ref(CollectingDataProcedures): Collecting data {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CollectSurveyQuestions1).**
No place for $18$-year-olds.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CollectSurveyQuestions1B).**
'$2$ children' in two categories (not mutually exclusive); '$4$ children' does not belong in any category (not exhaustive).
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CollectSurveyQuestions2).**
Best: second.
First: *leading* (*concerned* cat owners...)
Third: *leading* (Do you *agree*...)
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CollectSurveyQuestions2B).**
First: *leading* ('environmentally friendly').
First, second: only present the option of 'owning'
Third: best question.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SunscreenQuestions).**
First fine; 'seldom' (for instance) may have different meanings to different people; possible recall bias.
Second: overlapping options (both $1\hs$ and $2\hs$ in two categories).
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:KidsEnvironmentQuestions).**
First two: *closed* (one option to select). 
Other *open* (produces qualitative data).
Not sure primary school children could recall answers to first $2$\ questions.
:::
`r if( !includeEvenAnswers) "-->"`

::::::







\pagebreak <!--               MANUAL           -->








### Chap.\ \@ref(DescribingVars): Classifying data and variables {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifying1).**
Quant.\ continuous.
Qual.\ nominal.
Quant.\ continuous.
Qual.\ nominal.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifying1B).**
Quant.\ discrete.
Qual.\ nominal.
Qual.\ nominal.
Qual.\ nominal.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifying2).**
F, T, F.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifying2B).**
False; false; true.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifying3).**
Nominal; qualitative.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifyingGraphsLimeTrees).**
*Foliage biomass*: quant.\ continuous.
*Tree diameter* (cm): quant.\ continuous.
*Age of the tree* (years): quant.\ continuous.
*Origin of the tree*: qual.\ nominal.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:VariablesLevelsA).**
Sex of person
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:VariablesLevelsB).**
Whether woman aged under\ $40$.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifyingVariables1).**
**1.** Quant.\ continuous.
**2.** Qual.\ nominal.
**3.** Qual.\ ordinal.
**4.** Quant.\ discrete.
:::

`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifyingVariables2).**
**1.** Qual.\ ordinal.
**2.** Qual.\ nominal.
**3.** Qual.\ nominal.
**4.** Qual.\ ordinal.
**5.** Quant.\ continuous.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:BRFSS).**
**1.** Qual.\ nominal.
**2.** Quant.\ discrete.
**3.** Qual.\ ordinal (perhaps quant.\ discrete).
**4.** Qual.\ nominal.
**5.** Quant.\ continuous.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NHANES).**
**1.** Qual.\ ordinal.
**2.** Quant.\ discrete
**3.** Quant.\ discrete.
**4.** Qual.\ nominal.
**5.** Qual.\ ordinal.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifyingOrthoses).**
*Gender*: qual.\ nominal.
*Age*: quant.\ continuous.
*Height*: quant.\ continuous.
*Weight*: quant.\ continuous.
*<span style="font-variant:small-caps;">gmfcs</span>*: qual.\ ordinal.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifyingNitrogenInSoil).**
*Fertiliser dose*: quant.\ continuous.
*Soil nitrogen*: quant.\ continuous.
*Fertiliser source*: qual.\ nominal.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeClassifyingKangaroos).**
*Kangaroo response*: qual.\ ordinal (perhaps nominal?).
*Drone height*: quant.; four values used; probably treated as qual.\ ordinal.
*Mob size*: quant.\ discrete.
*Sex*: qual.\ nominal.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DescribeSelfieDeaths).**
*Location* only variable (observed from *individuals*).
*Number people* and *percentage people* who died is *summary* of data from individuals.
'Location': *nominal*, *qualitative* (seven *levels*).
:::
`r if( !includeEvenAnswers) "-->"`

::::::





### Chap.\ \@ref(SummariseQuantData): Summarising quantitative data {.unlisted .unnumbered}


:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphABSdeaths).**
Shape: skewed *left*.
Average: perhaps $70$--$100$?
Variation: most between $30$, $80$.
Outliers: none; 'bump' at lower ages.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphLimeTrees).**
Very right-skewed.
Average near $1$ or $2\kgs$.
Most between\ $0$ and $5\kgs$.
Possible outlier near $11\kgs$.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantNHANES).**
**1.** Probably median (but mean probably OK).
**2.** Slightly right skewed; average near $1.5\mmols$/L; most between\ $3$, $4\mmols$/L; some large outliers.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantCherryRipes).**
Maybe quote *both* mean and median ($14.90$ and\ $14.99\gs$ respectively).
Average around $15.0\gs$.
Most between\ $13.5$ and\ $16.0\gs$.
Slightly skewed left, perhaps one low outlier.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantRides).**
<!-- rideFatalities <- c(2,4,3,4,7,6,1,3,2,5) -->
**1.** $3.7$.
**2.** $3.5$.
**3.** $1.888562$.
**4.** $5 - 2 = 3$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantFulmars).**
**2.** $643\gs$.
**3.** $12.884\gs$.
**4.** $637.5\gs$.
**5.** We do not know.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantSOI).**
<!-- SOI <- c(0.8, 4.6, -19.8, 9.8, 2.1, 5.3, -8.2, -14.6, -1.8) -->
Plot not shown.
**1.** $-2.42$.
**2.** $0.8$.
**3.** $29.6$ ($-19.8$ to\ $9.8$).
**4.** $9.831172$; about $9.83$.
**5.** IQR: $4.95 - (-11.4) = 16.35$ (*not* including median in each half).
(No units of measurement.)
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WeightFries).**
**1.** Not shown.
**2.** $\bar{x} = 144.3\gs$; median: $143.3\gs$; $s = 11.61\gs$; IQR: $16.25\gs$.
**3.** Unlikely (but only one of many samples).
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SummaryQuantOrthoses).**
**1.** In cm: $127.4$; $129.0$; $14.4$; $24$ from software.
       Manually (*without* median in each half): $Q_1 = 113$, $Q_3 = 138$, IQR is $25$.
**2.** Don't know.
**3.--5.** Not shown.
**6.** Hard to describe with standard language; approx. symmetric?.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantMatchingMicroPlastics).**
**2.** $\bar{x} = 3.09$; median: $2.0$.
**3.** Don't know.
**4.** $s = 2.77$.
**5.** IQR: $4$.
:::
`r if( !includeEvenAnswers) "-->"`


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantMatchingMicroPlastics).**
*Average*: tricky; maybe between $10$ or\ $15$.
*Variation*: about $0$ to\ $40$. 
*Shape*: slightly skewed right. 
*Outliers*: no outliers or unusual observations; observation between $35$ and\ $40$ *may* be an outlier.
  Probably not; a larger sample may have observations between $30$ and\ $35$.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantDescribeBrainFreezeHistogram).**
In seconds: Average about $10$?; variation $0$ to\ $30$ perhaps; skewed right.
Value between $35$ and\ $40$ perhaps outlier.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantCompareSD).**
Dataset\ B: more observations near mean; the average distance from mean smaller.
Standard deviation for Dataset\ B **smaller** than for Dataset\ A.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:JeansIQR1).**
**1.** Men's: about $50$%; women's: about $100$%. 
**2.** Men's: about  $0$%; women's: about $50$%.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:JeansIQR2).**
@PuddingJeans recorded data on the price of different styles of women's jeans (Fig.\ \@ref(fig:JeansBoxplot), right panel).

**1.** About $75$%.
**2.** About $50$%.
**3.** About $50$%.
**4.** Boot-cut.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:StatsHistograms).**
D; C; A; D.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:StatsHistograms2).**
D; C; B; D; C.
:::
`r if( !includeEvenAnswers) "-->"`


::::::



```{r eval=FALSE, include=FALSE}
Fatalities <- c(2, 4, 3, 4, 7, 6, 1, 3, 2, 5)
mean(Fatalities)
median(Fatalities)
sd(Fatalities)
```


```{r eval=FALSE, include=FALSE}
Fulmars <- c(635, 635, 668, 640, 645, 635)
mean(Fulmars)
sd(Fulmars)
median(Fulmars)
```


```{r include=FALSE}
data(Orthoses)

mean(Orthoses$Height)
median(Orthoses$Height)
sd(Orthoses$Height)
IQR(Orthoses$Height)

#hist(Orthoses$Height)
```

```{r eval=FALSE, include=FALSE}
SOIvals <- c(0.8, 4.6, -19.8, 9.8, 2.1, 5.3, -8.2, -14.6, -1.8)

mean(SOIvals)
median(SOIvals)
range(SOIvals)
sd(SOIvals)
IQR(SOIvals)
```



```{r eval=FALSE, include=FALSE}
Microplastics <- c(8, 3, 5, 2, 0, 2, 1, 7, 5, 1, 0)
mean(Microplastics)
median(Microplastics)
sd(Microplastics)
IQR(Microplastics)
```

```{r WeightFriesHisto, eval=includeGraphsTables, fig.width=4, fig.height=3, out.width='40%', fig.align="center", fig.cap="Histogram of weights of orders of large fries"}
data(FriesWt)
hist(FriesWt$FriesWt,
     las = 1,
     main = "Weight of large fries\n(Target: 171 g)",
     ylab = "Number of orders",
     xlab = "Weight of large fries (in g)")
abline(v = 171)
```



### Chap.\ \@ref(SummariseQualData): Summarising qualitative data {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SpiderMonkeys).**
**1.** Graph not shown; no commonly-observed social group include\ M.
**2.** Mode: many F plus offspring; median inappropriate.
<!-- See Fig.\ \@ref(fig:SpiderMonkeys).-->
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:QualSummary).**
**1.** *Where*: nominal (respondents can select more than one option; each option a Yes/No variable).
       *Temp*: quantitative, but three options: treat as ordinal.
       *Time*: ordinal.
**2.** Not shown.
**3.** *Where*: home, work most common (modes).
       *Temp*: $100$^o^C is mode, median among those who know.
       *Time*: $3\mins$ is mode, median among those who know.
       Counts can be turned to percentages and odds.
**4.** *Where*: mode is 'Home'. 
       *Temp* (ignoring unknowns): mode and median is $100$^o^C.
       *Time* (ignoring unknowns): mode and median is About $3\mins$.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphsCars).**
None *bad*.
I'd prefer bar chart; any OK.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphSurveyData).**
Bar (or dot) chart.
Pie chart inappropriate: more than one option can be selected.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OrdinalMedians).**
**1.** Gender: nominal; others ordinal.
**2.**--**4.** 
Gender: modes are F, M; no median.
Place: mode is city $> 100\,000$ residents; median is city $20\,000$ to\ $100\,000$ residents
Response: mode is 'Agree'; median is 'Neutral.
**5.** $5.12$: respondents about $5$\ times more likely to come from city than rural.
**6.** $0.613$: respondents about $0.61$\ times as likely to agree, strongly disagree than choose other option.
**7.** $1$: respondents as likely to be\ M as\ F.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ReclaimedWater).**
Plots not shown.
**4.** Advantage: availability. Disadvantage: high price.
**5.** Not shown.
**6.** $0.209$.
**7.** $1.21$.
**8.** First includes farmers not identifying disadvantages. 
:::
`r if( !includeEvenAnswers) "-->"`
<!-- See Fig.\ \@ref(fig:ReclaimedWaterPlots) for some plots.-->






::: {.answer data-latex=""}
**Ex.\ \@ref(exr:StudentTransport).**
**1.** Walking; Bus
**2.** Bus.
**3.** No.
**4.** $0.102$, $0.123$; $0.246$, $0.412$, $0.116$.
**5.** $3.44$; i.e., students $3.44$ times as likely to use motorised transport than active.
**6.** $0.141$; i.e, for every $100$ students that *do not walk*, $100\times 0.141 = 14.1$ *do walk*.
**7.** Not shown.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:QualSumBabyBoom).**
**1.** M.
**2.** Medians not appropriate.
**3.** $40.9$% F, $59.1$ M.
**4.** $0.692$; a birth less likely to be F than\ M.
**5.** Not shown.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:FEVplots).**
**1.** Not shown.
**2.** Gender: M; smoking: Non-smoking.
**3.** Gender: percentage, odds M: $51.4$%, $1.06$; smoking: percentage, odds non-smoking: $9.9$%, $0.11$.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SummariseUniOrthoses).
*Gender*: bar or dot: mode: M. F: $20$%; odds: $0.25$.
*MFCS*: bar or dot: mode: Level\ I $80$%; odds: $4$.
*Age*, *Height*, *Weight*: histogram.
:::
`r if( !includeEvenAnswers) "-->"`


::::::


```{r TransportPlots, eval=includeGraphsTables, fig.align="center", fig.cap="Methods of transport to campus. White refers to active methods of transport; shaded to motorised methods of transport.", out.width='90%', fig.width=8, fig.height=3}
TransportTable <- array( dim = c(5, 2) )

rownames(TransportTable) <- c("Bicycle",
                              "Walking",
                              "Car",
                              "Bus",
                              "Other")
colnames(TransportTable) <- c("Number",
                              "Percentage")
TransportTable[, 1] <- c(29, 35, 70, 117, 33)
TransportTable[, 2] <- c(10.2, 12.3, 24.7, 41.2, 11.6)


par( mfrow = c(1, 2) )

par( mar = c(3.1, 5.5, 4.1, 2.25) )

barplot( TransportTable[, 1],
         las = 1,
         xlim = c(0, 120),
         horiz = TRUE,
         xlab = "Number of students",
         ylab = "",
         main = "Methods of transport",
         col = c("white",
                 "white",
                 plot.colour,
                 plot.colour,
                 plot.colour) )
box()

###

Mode <- c( Active    = sum(TransportTable[1:2, 1]),
           Motorised = sum(TransportTable[3:5, 1]))
pie( Mode,
     xlim = c(-2, 2),
     ylim = c(-2, 2),
     col = c("white",
             plot.colour),
     main = "Methods of\ntransport")
```



```{r SpiderMonkeys, eval=includeGraphsTables, fig.cap="Dot chart of spider monkey family groups", fig.align="center", fig.width=5.5, fig.height=3.5}
counts <- c(8, 3, 2, 15, 1, 23, 48)
names <- c("Solitary",
           "All males",
           "Female + no young",
           "Mixed young",
           "Mixed + no young",
           "One female + offspring",
           "Many females + offspring")

sort.order <- sort( counts,
                    index.return = TRUE)

old.mar = par()$mar
new.mar <- old.mar
new.mar[2] <- old.mar[2] * 2.75
par(mar = new.mar )

dotchart( counts[sort.order$ix],
          pch = 19,
          las = 1,
          xlim = c(0, 50),
          xlab = "Percentage of observations",
          ylab = "",
          main = "Spider monkey sub-groups")
axis(side = 2,
     seq_along(counts[ sort.order$ix ]),
     names[sort.order$ix],
     las = 1)
abline(v = 0,
       col = "grey")   # Add zero line for clarity

```




```{r , eval=includeGraphsTables}
Advantages <- c(15, 27, 16)
namesAdvantages <- c("Water reutilization",
                     "Availabiity",
                     "Sustainability")
Disadvantages <- c(40, 12, 21)
namesDisadvantages <- c("High price",
                        "Growing conductivity",
                        "Lack of proper filtering")
AdTable <- DisadTable <- array( dim = c(3, 2))

AdTable[, 1] <- namesAdvantages
AdTable[, 2] <- Advantages
DisadTable[, 1] <- namesDisadvantages
DisadTable[, 2] <- Disadvantages


AdTable[, 2] <- round(Advantages/sum(Advantages) * 100, 1)
DisadTable[, 2] <- round(Disadvantages/sum(Disadvantages) * 100, 1)

if( knitr::is_latex_output() ) {
T1 <-  knitr::kable(pad(AdTable,
                        surroundMaths = TRUE,
                        targetLength = c(0, 2),
                        decDigits = 0),
                    format = "latex",
                    valign = 't',
                    align = c("r", "c"),
                    linesep = "",
                    col.names = c("Advantage", "No. farmers"),
                    row.names = FALSE,
                    escape = FALSE,
                    booktabs = TRUE) %>%
  row_spec(0, bold = TRUE)


T2 <-  knitr::kable(pad(DisadTable,
                        surroundMaths = TRUE,
                        targetLength = c(0, 2),
                        decDigits = 0),
                    format = "latex",
                    valign = 't',
                    align = c("r", "c"),
                    linesep = "",
                    col.names = c("Disadvantage", "No. farmers"),
                    row.names = FALSE,
                    escape = FALSE,
                    booktabs = TRUE) %>%
  row_spec(0, bold = TRUE)



out <- knitr::kables( list(T1, T2),
                      format = "latex",
                      label = "ReclaimedWaterPercent",
                      caption = "The advantages and disadvantages of using reclaimed water, reported by $231$ farmers.") %>%
  kable_styling(font_size = 7)
out2 <- prepareSideBySideTable(out)
out2
}


if( knitr::is_html_output() ) {
  T1 <-  knitr::kable(pad(AdTable,
                        surroundMaths = TRUE,
                        targetLength = c(0, 2),
                        decDigits = 0),
                    format = "html",
                    valign = 't',
                    align = c("r", "c"),
                    linesep = "",
                    col.names = c("Advantage", "No. farmers"),
                    row.names = FALSE,
                    escape = FALSE,
                    booktabs = TRUE) %>%
  row_spec(0, bold = TRUE)


T2 <-  knitr::kable(pad(DisadTable,
                        surroundMaths = TRUE,
                        targetLength = c(0, 2),
                        decDigits = 0),
                    format = "html",
                    valign = 't',
                    align = c("r", "c"),
                    linesep = "",
                    col.names = c("Disadvantage", "No. farmers"),
                    row.names = FALSE,
                    escape = FALSE,
                    booktabs = TRUE) %>%
  row_spec(0, bold = TRUE)



knitr::kables( list(T1, T2),
                      format = "html",
                      label = "ReclaimedWaterPercent",
                      caption = "The advantages and disadvantages of using reclaimed water, reported by $231$ farmers.") %>%
  kable_styling(font_size = 7)
}
```




```{r, eval=includeGraphsTables, fig.width=7, fig.width=7, out.width='90%', ReclaimedWaterPlots, fig.align="center", fig.cap="Plots for the water-reclaiming exercise"}
par( mfrow = c(2, 2))
barplot(Advantages, 
        main = "Advantages",
        las = 1)
pie(Advantages,
    main = "Disadvantages")

barplot(Disadvantages,
        las = 1)
dotchart(Disadvantages, 
         main = "Disadvantages",
         las = 1, 
         xlim = c(0, 40))
```





### Chap.\ \@ref(SummariseWithin): Comparing quantitative data within individuals {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CompareWithinInsulation).**
**1.** House.
**2.** Each house has before, after.
Graph, table not shown.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CompareWithinExercisesCaptopril).**
No answer (yet).
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CompareWithinPainRelief).**
Graph, table not shown.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CompareWithinStressSurgeryCI).**
**1.** Each patient has $2$ measurements,
**2.** Std\ dev.\ for increase found from differences.
Graph not shown.
:::
`r if( !includeEvenAnswers) "-->"`


<!-- ::: {.answer data-latex=""} -->
<!-- **Ex.\ \@ref(exr:CompareWithinCOVID).** -->
<!-- No answer (yet). -->
<!-- ::: -->


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CompareWithinRunning).**
Not shown.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CompareWithinWCTennis).**
**1.** How much longer push-time is with racquet.
Graph and table not shown.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CompareWithinJumping).**
**1.** How much further people jump in shoes.
Graph and table not shown.
:::



::::::





### Chap.\ \@ref(BetweenQuantData): Comparing quantitative data between individuals {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:BoxplotsProjectCosts).**
**1.** In general, DB smaller cost over-runs.
**2.** Tricky: DB: $2\cms$; DBB: $3\cms$.
**3.** Tricky: DB: $2\cms$; DBB: $3\cms$. 
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphsAIS).**
**1.** In general, basketballers taller than netballers.
**2.** B: About $185\cms$; N: about $175\cms$.
**3.** B: about $14\cms$; N: about $6\cms$.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantMatchingHistogramsAndBoxplots).**
I: B (mean; standard deviation).
II: A (median; IQR).
III: C (median; IQR).
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalJellyfish).**
**1.** Neither correct: smallest jellyfish $6\cms$ breadth.
**2.** Sample small, hard to be certain.
Average: about $10\mms$; data from about $6$\ to\ $16\mms$; slightly right skewed; no outliers.
**3.** Site\ B generally larger.
**4.** Site\ A: about $10\mms$; Site\ B: about $17\mms$. 
**5.** Site\ A.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQuantConstructionWorkerProductivity).**
**1.** $0.61$; $0.40$; $0.42$ panels/min.
**4.** Worker\ 2 faster, more consistent (using IQR); Worker\ 1 slower.
Plots not shown.
:::



`r if( !includeEvenAnswers) "<!--"`
:::{.answer data-latex=""}
**Ex.\ \@ref(exr:GreenBuilding).**
**1.** $2.7$^o^C; $3.1$^o^C; $2.6$^o^C.\spacex
**4.** Office\ A cooler.
Plots not shown.
:::
`r if( !includeEvenAnswers) "-->"`
<!-- See Fig.\ \@ref(fig:OfficeTempsBoxplot). -->




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CompareQuantExercisesNHANES).**
**1.** Error bar chart.
**2.** Not shown.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:FacePlantSummary).**
No answer (yet).
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:QuantCompareSpeedSignage).**
**1.** Not shown.
<!-- Table\ \@ref(tab:SignageSummary). -->
**2.** Not shown.
<!-- Fig.\ \@ref(fig:SpeedBoxplot).-->
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CompareQuantDeceleration).**
**1.** Not shown.
**2.** Not shown.
**3.** Acceleration.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:QuantCompareTyping).**
**1.** `mAcc`: highly *left* skewed; `Age`: highly *right* skewed; `mTS`: slightly right skewed.
Perhaps medians, IQRs for summarising (mean, std dev.\ probably OK for `mTS`).
<!-- See Table\ \@ref(tab:TypingUnivar). -->
**2.** Not shown.
<!-- See Table\ \@ref(tab:TypingCompare). -->
**3.** Little diff between M, F in *sample*.
:::




`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:QuantCompareDental).**
No answers (yet).
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:QuantCompareSnakesConfounding).**
**1.** Very similar mean SVL.
**2.** Crayfish regions: smaller mean SVL.
**3.** Crayfish regions: larger mean SVL.
**4.** Confounding.
:::

::::::





```{r PanelsBoxplot, eval=includeGraphsTables, fig.align="center", fig.cap="The boxplot for the productivity data", fig.height=3, fig.width=5, include=FALSE}
Panels <- array(NA, 
                dim = c(7, 3))

rownames(Panels) <- c("Mean", 
                      "Minimum", 
                      "1st quartile", 
                      "Median", 
                      "3rd quartile", 
                      "Maximum", 
                      "Range")
colnames(Panels) <- c("Worker 1", 
                      "Worker 2", 
                      "Worker 3")

Panels[, 1] <- c(1.24, 0.59, 0.88, 1.35, 1.49, 1.88, 1.28)
Panels[, 2] <- c(1.73, 1.13, 1.51, 1.70, 1.91, 3.00, 1.87)
Panels[, 3] <- c(1.36, 0.86, 1.16, 1.38, 1.58, 2.17, 1.31)

Panels.bxp <- list(
  stats = as.matrix(Panels[2:6, ]),
  n = rep(10, 3), # No idea; a guess
  conf = matrix( NA, ncol = 3),
  out = rep( numeric(0), 3),
  group = rep(numeric(0), 3),
  names = c("Worker 1", 
            "Worker 2", 
            "Worker 3")
)


bxp( Panels.bxp, 
     las = 1,
     ylab = "Productivity (panels/minute)")
```



```{r AVstackedbar, eval=includeGraphsTables, out.width='70%', fig.width=6, fig.height = 3.5, fig.align="center", fig.cap="A stacked bar chart of the AV data"}
Driving <- c(58, 79, 96, 97, 70)
Cycling <- c(77, 104, 87, 76, 56)
Walking <- c(63, 86, 103, 82, 66)

AV2 <- cbind(
  Driving = Driving,
  Cycling = Cycling,
  Walking = Walking)

par( mar = c(4.5, 4, 6, 2) + 0.1 )
barplot(AV2,
        ylim = c(0, 4.8),
        horiz = TRUE,
        col = viridis(8)[2:6],
        las = 1,
        main = "Perceptions of safety near AVs",
        xlab = "Responses")
text( x = c(25, 110, 200, 300, 375),
      y = 4.3,
      cex = 0.9,
      labels = c("Unsafe",
                 "Somewhat\nunsafe",
                 "Neutral",
                 "Somewhat\nsafe",
                 "Safe") )

```



```{r SignageSummaryDec, eval=includeGraphsTables}
data(Deceleration)

SignageSummaryDec <- array(dim = c(3, 4))
rownames(SignageSummaryDec) <- c("Before",
                                 "After",
                                 "Change")
colnames(SignageSummaryDec) <- c("Mean",
                                 "Std deviation",
                                 "Std error",
                                 "Sample size")

Speed.n <- tapply(Deceleration$Deceleration,
                  list(Deceleration$When),
                  "length")
Speed.sd <- tapply(Deceleration$Deceleration,
                   list(Deceleration$When),
                   "sd")

SignageSummaryDec[1:2, 1] <- round( tapply(Deceleration$Deceleration,
                                           list(Deceleration$When),
                                           "mean"),
                                    4)
SignageSummaryDec[1:2, 2] <- round( Speed.sd, 4)
SignageSummaryDec[1:2, 3] <- round( Speed.sd / sqrt(Speed.n), 5)
SignageSummaryDec[1:2, 4] <- Speed.n

SignageSummaryDec[3, 1] <- SignageSummaryDec[1, 1] -
  SignageSummaryDec[2, 1]

sdB <- round( SignageSummaryDec[1, 3], 4)
sdA <- round( SignageSummaryDec[2, 3], 4)
SignageSummaryDec[3, 3] <- round( sqrt( sdB^2/Speed.n[1] +
                                          sdA^2/Speed.n[2] ),
                                  5)

SignageSummaryDec[3, 4] <- NA
SignageSummaryDec[3, 2] <- NA



if( knitr::is_latex_output() ) {
  kable(pad(SignageSummaryDec,
            surroundMaths = TRUE,
            targetLength = c(6, 6, 7, 2),
            decDigits = c(4, 4, 5, 0)),
        format = "latex",
        longtable = FALSE,
        booktabs = TRUE,
        escape = FALSE,
        align = "c",
        col.names = c("Mean",
                      "Std deviation",
                      "Std error",
                      "Sample size"),
        caption = "The signage data summary (in m/s-squared)"
  ) %>%
    row_spec(0, bold = TRUE) %>%
    row_spec(3, italic = TRUE) %>%
    kable_styling(font_size = 7)
}
if( knitr::is_html_output() ) {
  kable(pad(SignageSummaryDec,
            surroundMaths = TRUE,
            targetLength = c(6, 6, 7, 2),
            decDigits = c(4, 4, 5, 0)),
        format = "html",
        longtable = FALSE,
        booktabs = TRUE,
        align = "c",
        col.names = c("Mean",
                      "Std deviation",
                      "Std error",
                      "Sample size"),
        caption = "The signage data summary (in m/s-squared)" ) %>%
    kable_styling(full_width = FALSE)
}
```



```{r TypingUnivar, eval=includeGraphsTables}
data(Typing)

TypingSummary <- array( dim = c(5, 3) )

colnames(TypingSummary) <- c("Typing accuracy (percent)",
                             "Typing speed (wpm)",
                             "Age (years)")

rownames(TypingSummary) <- c("Median",
                             "IQR",
                             "Shape",
                             "Outliers",
                             "Sample size")

TypingSummary[1, ] <-   c( median(Typing$mAcc, na.rm = TRUE),
                           median(Typing$mTS, na.rm = TRUE),
                           median(Typing$Age, na.rm = TRUE) )
TypingSummary[2, ] <-   c( IQR(Typing$mAcc, na.rm = TRUE),
                           IQR(Typing$mTS, na.rm = TRUE),
                           IQR(Typing$Age, na.rm = TRUE) )
TypingSummary[3, ] <- c("Left skewed",
                        "Slight right",
                        "Right skewed")
TypingSummary[4, ] <- c("Some small",
                        "Some large",
                        "Some large")
TypingSummary[5, ] <- c( paste0("$", realLength(Typing$mAcc, na.rm = TRUE), "$"),
                         paste0("$", realLength(Typing$mTS, na.rm = TRUE), "$"),
                         paste0("$", realLength(Typing$Age, na.rm = TRUE), "$") )

if( knitr::is_latex_output() ) {
  kable(pad(TypingSummary,
            surroundMaths = TRUE,
            targetLength = c(5, 5, 2),
            decDigits = c(3, 3, 0)),
        format = "latex",
        longtable = FALSE,
        booktabs = TRUE,
        align = "c",
        col.names = c("(in percent)",
                      "(in wpm)",
                      "(in years)"),
        escape = FALSE,
        caption = "Numerical summary of the typing data") %>%
    row_spec(0, bold = TRUE) %>%
    column_spec(1, bold = TRUE) %>%
    kable_styling(font_size = 7) %>%
    add_header_above( c(" " = 1,
                        "Typing accuracy" = 1,
                        "Typing speed" = 1,
                        "Age" = 1),
                      bold = TRUE,
                      line = FALSE)
}
if( knitr::is_html_output() ) {
  kable(pad(TypingSummary,
            surroundMaths = TRUE,
            targetLength = c(5, 5, 2),
            decDigits = c(3, 3, 0)),
        format = "html",
        longtable = FALSE,
        booktabs = TRUE,
        align = "c",
        escape = FALSE,
        caption = "Numerical summary of the typing data") %>%
    kable_styling(full_width = FALSE) %>%
    row_spec(0, bold = TRUE) %>%
    column_spec(1, bold = TRUE)
}
```


```{r TypingCompare, eval=includeGraphsTables}
TypingSummary <- array( dim = c(5, 6) )

colnames(TypingSummary) <- c("Males",
                             "Females",
                             "Males",
                             "Females",
                             "Males",
                             "Females")

rownames(TypingSummary) <- c("Median",
                             "IQR",
                             "Shape",
                             "Outliers",
                             "Sample size")

TypingSummary[1, ] <-   c( tapply(Typing$mAcc, list(Typing$Sex), "median", na.rm=TRUE),
                           tapply(Typing$mTS, list(Typing$Sex), "median", na.rm=TRUE),
                           tapply(Typing$Age, list(Typing$Sex), "median", na.rm=TRUE) )
TypingSummary[2, ] <-   c( tapply(Typing$mAcc, list(Typing$Sex), "IQR", na.rm=TRUE),
                           tapply(Typing$mTS, list(Typing$Sex), "IQR", na.rm=TRUE),
                           tapply(Typing$Age, list(Typing$Sex), "IQR", na.rm=TRUE) )
TypingSummary[3, ] <- c("L. skewed",
                        "L. skewed",
                        "Slight R.",
                        "Slight R.",
                        "R. skewed",
                        "R. skewed")
TypingSummary[4, ] <- c("Some small",
                        "Some small",
                        "Some large",
                        "Some large",
                        "Some large",
                        "Some large")
TypingSummary[5, ] <- c( paste0("$", tapply(Typing$mAcc, list(Typing$Sex), "realLength"), "$"),
                         paste0("$", tapply(Typing$mTS, list(Typing$Sex), "realLength"), "$"),
                         paste0("$", tapply(Typing$Age, list(Typing$Sex), "realLength"), "$") )

if( knitr::is_latex_output() ) {
  kable(pad(TypingSummary,
            surroundMaths = TRUE,
            targetLength = 5,
            decDigits = c(3, 3,
                       2, 2,
                       1, 1,
                       0, 0)),
        format = "latex",
        longtable = FALSE,
        booktabs = TRUE,
        align = "c",
        escape = FALSE,
        col.names = NA,
        caption = "Numerical summary of the typing data") %>%
    row_spec(0, bold = TRUE) %>%
    column_spec(1, bold = TRUE) %>%
    kable_styling(font_size = 7) %>%
    add_header_above( c(" " = 1,
                        "Typing accuracy" = 2,
                        "Typing speed" = 2,
                        "Age" = 2),
                      bold = TRUE,
                      line = TRUE)

}
if( knitr::is_html_output() ) {
  kable(pad(TypingSummary,
            surroundMaths = TRUE,
            targetLength = 5,
            decDigits = c(3, 3,
                       2, 2,
                       1, 1,
                       0, 0)),
        format = "html",
        longtable = FALSE,
        booktabs = TRUE,
        align = "c",
        escape = FALSE,
        caption = "Numerical summary of the typing data") %>%
    kable_styling(full_width = FALSE) %>%
    row_spec(0, bold = TRUE) %>%
    column_spec(1, bold = TRUE)
}
```


```{r OfficeTempsBoxplot,  eval=includeGraphsTables, fig.cap="Boxplot of the office temperatures", fig.align="center", fig.width=5, fig.height=3}
Green.stats <- matrix(
 c(16.4    ,15.9    ,20.1,
   22.8    ,23.8    ,24.6,
    24.4    ,25.5    ,26.1,
    25.5   ,26.9    ,27.2,
    27.4    ,31.0    ,30.3),
  ncol = 3,
  byrow = TRUE)

Green.bxp <- list( stats = Green.stats,
                   n = rep(100, 3), # MADE UP
                   conf = NA,
                   out = rep(numeric(0), 3),
                   group = rep(numeric(0), 3),
                   names = c("Office A",
                             "Office B",
                             "Office C"))
bxp( Green.bxp,
     las = 1,
     xlab = "Office",
     ylab = "Temperature\n(in degrees C)",
     ylim = c(15, 35)
)
```




```{r SpeedBoxplot, eval=includeGraphsTables, fig.cap="Boxplot of speed before and after adding signage", fig.align="center", out.width='50%', fig.width=5, fig.height=3.25  }

data(Speed) ### Exercise

Speed$When <- factor(Speed$When,
                     levels = c("Before", "After"),
                     ordered = TRUE)

Speed2 <- data.frame( Before = Speed$Speed[ Speed$When == "Before" ][1:10],
                      After = Speed$Speed[ Speed$When == "After" ][1:10])

par(mar = c(5, 4, 4, 4) + 0.1)
boxplot( Speed2,
        col = plot.colour,
        las = 1,
        ylab = "Speed (in km/h)",
        xlab = "When speed measured",
        main = "Speeds before and after adding speed signage",
          names = c("", "") )
 abline( h = 100,
         lwd = 2)
 axis(side = 4,
      at = 100,
      las = 1,
      labels = "Speed\nlimit")
 mtext(text = c(expression( atop(Before, group("(",italic(n) == 38, ")")) ),
                expression( atop(After, group("(", italic(n) == 41, ")")) ) ),
       side = 1,
       at = 1:2,
       las = 1,
       line = 2 )
```



```{r CPalsyPlots, eval=includeGraphsTables, fig.align="center", fig.cap="Some graphs from the cerebral palsy data", fig.height=2.0, fig.width=7, include=FALSE, out.width='100%'}
NumP <- 15
Gender <- rep("M",
              NumP)

Gender[ c(11, 14, 15)] <- "F"
Age <- c(9, 7, 7, 12, 11, 5, 6, 8, 8, 6, 7, 11, 7, 9, 8)
Ht <- c(136, 106, 129, 152, 146, 113, 112, 112, 138, 116, 113, 141, 136, 128, 133)
Wt <- c(34.5, 16.2, 21.1, 40.4, 39.3, 18.1, 16.7, 19.1, 28.6, 19.3, 17.6, 34.9, 34.5, 21.9, 23.0)
GMFCS <- c(1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1)

CPalsy <- data.frame(Gender = Gender,
                     Age = Age,
                     Height = Ht,
                     Weight = Wt,
                     GMFCS = GMFCS)

par( mfrow = c(1, 3))
plot( Height ~ Age,
      data = CPalsy,
      las = 1,
      xlab = "Age (in years)",
      ylab = "Ht (in cm)")

boxplot( Height ~ Gender,
         data = CPalsy,
         las = 1,
         xlab = "Gender",
         ylab = "Ht (in cm)")

barplot( xtabs( ~ GMFCS + Gender,
                data = CPalsy),
         las = 1,
         beside = FALSE,
         xlab = "Gender",
         ylab = "GMFCS")
```







### Chap.\ \@ref(CompareQualData): Comparing qualitative data between individuals {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQual1).**
Zero
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQual2).**
Negative.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:NumericalQualHangovers).**
**1.** 
*Vomited*: $0.50$ beer, wine; $0.50$ wine only.
*Didn't vomit*: $0.738$ beer, wine, $0.262$ wine only. 
Prop.\ drank various drinks, among those who did, didn't vomit.
**2.** 
*Beer, wine*: $8.8$% vomited, $91.2$% didn't.
*Wine only*: $21.4$% vomited, $78.6$% didn't. 
Percentage that vomited, for each drinking type.
**3.** $(6 + 6)/(6 + 6 + 62 + 22) = 0.125$.
**4.** $0.2727$.
**5.** $0.09677$.
**6.** $2.82$.
**7.** $0.354$.
**8.** $-0.176$.
**9.** Higher percentage vomited after beer+wine, compared to beer only.
:::




`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Wallabies).**
**2.** $0.32616$.
**3.** About $2.07$.
**4.** About $1.69$.
**5.** About $37.1$%.
**6.** $1.22$.
**7.** $0.0453$.
**10.** Percentage F among adults slightly higher than among young.
Tables, graph not shown.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OddsAugustRainfall).**
**1.** About $18.4$%.
**2.** About $25.9$%.
**3.** About $11.7$%.
**4.** About $0.226$.
**5.** $0.35$.
**6.** About $0.132$.
**7.** About $2.7$.
**8.** Odds no August rainfall in Emerald $2.7$ times higher in months with non-positive SOI.
:::




`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OddsBackpacks).**
**1.** $45.9$%.
**2.** $61.4$%.
**3.** $0.848$.
**4.** $1.59$.
**5.** $1.15$.
**6.** $0.533$.
**7.** Odds of reporting back pain from carrying school bags, comparing boys to girls.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:AVquestions).**
Not shown.
:::
<!-- See Fig.\ \@ref(fig:AVchart).-->




`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Roadkill).**
**1.** Season; sex.
**2.** Not shown.
**3.** Most cannot be sexed! Lower numbers in spring, summer
:::
`r if( !includeEvenAnswers) "-->"`





::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SkippingBreakfast).**
**1.** *Prop.*\ F skipped: $\hat{p}_F = 0.359$.
**2.** *Prop.*\ M skipped: $\hat{p}_M = 0.284$.
**3.** Odds(Skips breakfast, F): $0.5598$;
**4.** Odds(Skips breakfast, M): $0.3966$.
**5.** OR: $1.41$.
**6.** Odds F skipping $1.41$ *times* odds M skipping. 
**7.** Not shown. 
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CoffeeGreenTea).**
**1.** $13.2$%.
**2.** $2.3$%.
**3.** $0.152$.
**4.** $0.0238$.
**5.** $6.39$.
**6.** Odds coffee drinker being smoker: $6.39$ times odds of non-coffee drinker being smoker.
**7.** Not shown.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Dispatchers).**
**1.** Not shown.
**2.** $74.6%$.
**3.** $60.9$%.
**4.** $2.487$.
**5.** $1.558$.
**6.** $1.596$.
**7.** $0.626$.
**8.** Not shown.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SoccerHeaders).**
**1.** Student type: nominal ($3$ levels). 
       Number concussions: ordinal ($3$ levels).
**2.** $18.75$% of all college students in sample received $1$\ concussion.
**3.** Odds(non-athlete receiving $2+$ concussions): $0.35$.
       Non-athletes $0.35$ times as likely to receive $2+$ concussions than fewer than\ $2$. 
       Or: for every $100$ students with fewer than $2$\ concussions, $35$ with $2+$.
**4.** Odds(soccer player receiving $2+$ concussions): $0.26$.
**5.** OR: about $1.35$.
**6.** Not given.
**7.** Not given.
**8.** Probably row percentages.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:PLHomeAway).**
$\text{OR(W; home)} = 4/6  = 0.667$; $\text{OR(W; away)} = 7/4 = 1.75$.
$\text{OR} = 0.6667/1.75 = 0.381$.
:::

::::::



```{r, eval=includeGraphsTables}
AVtable2 <- array( dim = c(5, 3) )

colnames(AVtable2) <- c("Driving\nnear an AV",
                        "Cycling\nnear an AV",
                        "Walking\nnear an AV")
rownames(AVtable2) <- c("Unsafe",
                        "Somewhat unsafe",
                        "Neutral",
                        "Somewhat safe",
                        "Safe")

AVtable2[, 1] <- c(58, 79, 96, 97, 70)
AVtable2[, 2] <- c(77, 104, 87, 76, 56)
AVtable2[, 3] <- c(63, 86, 103, 82, 66)
```



```{r BreakSum, eval=includeGraphsTables}

Counts <- c(2383,
            1944,
            4257,
            4902)
Gender <- c("Females",
            "Males",
            "Females",
            "Males")
Breakfast <- c("Skip",
               "Skip",
               "Doesn't skip",
               "Doesn't skip")

Brek <- xtabs(Counts ~ Gender + Breakfast)[, c(2, 1)]
Brek2 <- cbind(Brek, "Total" = rowSums(Brek))


Brek.Sum <- array( NA,
                   dim = c(3, 3))
rownames(Brek.Sum) <- c("Females",
                        "Males",
                        "OR")

Brek.Sum[1, 1] <- Brek[1, 1] / sum(Brek[1,]) * 100
Brek.Sum[2, 1] <- Brek[2, 1] / sum(Brek[2,]) * 100

Brek.Sum[1, 2] <- Brek[1, 1] /Brek[1, 2]
Brek.Sum[2, 2] <- Brek[2, 1] /Brek[2, 2]

Brek.Sum[3, 2] <- Brek.Sum[1, 2] / Brek.Sum[2, 2]

Brek.Sum[1, 3] <- sum(Brek[1, ])
Brek.Sum[2, 3] <- sum(Brek[2, ])

if( knitr::is_latex_output() ) {
  kable(pad(Brek.Sum,
            surroundMaths = TRUE,
            targetLength = c(4, 5, 4),
            decDigits = c(1, 3, 0)),
        format = "latex",
        longtable = FALSE,
        booktabs = TRUE,
        escape = FALSE,
        align = c("r", "r", "r"),
        digits = c(1, 3),
        col.names = c("Percentage",
                      "Odds",
                      "Sample size"),
        caption = "Numerical summary of the Iranian-breakfast data: odds and percentage of those who skip breakfast") %>%
    row_spec(0, bold = TRUE) %>%
    kable_styling(font_size = 7) %>%
    row_spec(3, bold = TRUE) %>%
    row_spec(row = 2,
             hline_after = TRUE)
}
if( knitr::is_html_output() ) {
  kable(pad( Brek.Sum,
                surroundMaths = TRUE,
                targetLength = c(4, 5, 4),
                decDigits = c(1, 3, 0)),
           format = "html",
           longtable = FALSE,
           booktabs = TRUE,
           align = "c",
           digits = c(1, 3),
           col.names = c("Percentage",
                         "Odds",
                         "Sample size"),
           caption = "Numerical summary of the Iranian-breakfast data: odds and percentage of those who skip breakfast") %>%
    row_spec(3, bold = TRUE) %>%
    row_spec(row = 2,
             hline_after = TRUE)
}
```



```{r PossumRoadkillNumbers, , eval=includeGraphsTables, fig.align="center", fig.cap="Dot chart for the roadkill data", fig.height=4, fig.width=8, include=FALSE, out.width='80%'}

## Russell2009

Roadkill <- matrix( c(75, 25, 21,
                    74, 27, 22,
                    71, 10, 18,
                    58, 10, 12),
                  nrow = 4,
                  ncol = 3,
                  byrow = TRUE)
rownames(Roadkill) <- c("Autumn",
                        "Winter",
                        "Spring",
                        "Summer")
colnames(Roadkill) <- c("Unknown",
                        "Male",
                        "Female")

par( mar = c(5.1, 9.1, 4.1, 2.1))

dotchart( t(Roadkill),
        xlab = "Number",
        ylab = "",
        xlim = c(0, 80),
        pch = c(1, 2, 3),
        main = "Roadkill possums")
```




```{r, AVchart, eval=includeGraphsTables, fig.width = 7, fig.height = 4, fig.cap="Barchart for the AV data", out.width = '70%', fig.align = "center"}
par( mar = c(5.1, 9.1, 4.1, 2.1))

barplot(AVtable2,
        horiz = TRUE,
        las = 1,
        xlab = "Number",
        ylab = "",
        col = grey( seq(0, 1, length = 5)),
        ylim = c(0, 4.5),
        main = "Response to questions about AVs")

legend("top",
       ncol = 3,
       bty = "n",
       cex = 0.9,
       fill = grey( seq(0, 1, length = 5)),
       legend = rownames(AVtable2))

```





### Chap.\ \@ref(TwoQuant): Correlations between quantitative variables {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorrelationExerciseDrawNegR).**
Answers vary.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MatchCorrelations).**
You cannot be precise.
A: large; positive.
B: moderate; negative.
C: near zero.
D: not appropriate.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoQuantExercisesPeas).**
You cannot be precise.
Software: $r = 0.71$.
Realistically: 'reasonably high, positive $r$'.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoQuantExercisesPeas2).**
*Very* hard to estimate!
Software: $r = -0.13$.
Realistically: 'slightly negative'.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoQuantExercisesLimeTrees).**
**1.** A tree.
**2.** *Form*: starts straight-ish, then hard to describe.
       *Direction*: biomass increases as age increases (on average). 
       *Variation*: small-ish for small ages; large-ish for older trees (after $60$).
**3.** No.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoQuantExercisesBoneQuality).**
**1.** Person.
**2.** Approx. linear; negative; moderate?
**3.** Yes; approx. linear.
**4.** No: observational.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoQuantExercisesSoftdrink).**
**1.** Approximately linear; positive relationship; variation larger for more cases.
**2.** A delivery.
**3.** Non-constant variation: no.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoQuantExercisesMandible).**
Relationship prob. linear; some top-right observations different. 
Variation increases as Age increases.
Top right observations do not follow the linear relationship.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoQuantExercisesGorillas).**
No relationship.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoQuantExercisesWindmill).**
Non-linear; higher wind speed related to higher DC output (in general); small to moderate variation.
DC output increases as wind speed increases, not linearly.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SoilCN).**
Approx. linear; positive; strong.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:StudentWts).**
Approx. linear; positive; very strong.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoQuantExercisesONI).**
$R^2 = (-0.682)^2 = 0.465$: about $46.5$% of the unknown variation in number cyclones explained by knowing value of ONI.
:::

::::::




### Chap.\ \@ref(SummariseComments): More details about tables and graphs data {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Graphs123).**
Scatterplot; histogram of diffs; side-by-side bar.
:::


`r if( !includeEvenAnswers) "<!--"`
Histogram; histogram of diffs; boxplot.
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphsLimeTrees).**
Individual variables: *bar chart* for origin; *histogram* for others.
Between biomass, origin: *boxplot*.
Between biomass, other variables: *scatterplot*.
(On scatterplot, could encode origins with different colours, symbols.)
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphNitrogenInSoil).**
*Fertiliser* (quant.): histogram (response).
*Soil nitrogen* (quant.): histogram (explanatory).
*Source* (qual.\ nominal): bar chart (explanatory).
*Relationships*: between fertiliser dose, soil nitrogen: scatterplot. 
*Source* could be encoded using different coloured points.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphNoisyMiners).**
Plotting symbols unexplained.
Axis labels unhelpful.
Vertical axis could stop at $20$.
<!-- See Fig.\ \@ref(fig:MinersGoodCrabs) (left panel).-->
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphHorseshoeCrabs).**
Graph *inappropriate*: both variables *qualitative*.
Use stacked, side-by-side bar chart.
:::
`r if( !includeEvenAnswers) "-->"`
<!-- (Fig.\ \@ref(fig:MinersGoodCrabs), right panel).-->



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphsMADRS).**
**1.** Response: *change* in <span style="font-variant:small-caps;">madrs</span> (quant.\ cont.).
**2.** Explanatory: treatment group (qual.\ nominal, $3$\ levels).
**3.** Response: histogram. Explanatory: bar chart. Relationship: boxplot.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphsSkewBar).**
Variable is 'Sport' (*qual.\ nominal*).
Bars can be ordered any way.
*Skewness nonsense* (only for *quant.*\ variables).
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphsTyping).**
Plots not shown.
<!-- Some plots in Fig.\ \@ref(fig:TypingPlot). -->
*Speed*: average: around $60$\ wpm; variation: about $30$ to\ $120$\ wpm. 
Slightly right skewed; no obvious outliers.
*Accuracy*: average: around $85$%; variation: about $65$% to\ $95$%.
Left skewed; no obvious outliers.
*Age*: average: $25$; variation: about $15$ to\ $35$.
*Very* right skewed, perhaps unseen large outliers.
*Sex*: about twice as many F as M.
*Speed* and *Sex*: not big difference between M, F.\spacex
*Accuracy* and *Age*: hard to see relationship; no older people very slow.

Average speed, accuracy vary by age, not sex.
How data collected (self-reported? Or measured how?). 
How students obtained: a random, somewhat representative or self-selecting sample?
:::




`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteExercisesNHANES).**
*Graph*: black box hides median; vertical axis label unhelpful; horizontal axis unlabelled; units of measurement not given; title and/or caption unhelpful.

*Table*: units of measurement not given; no caption, or explanation; number decimal places inconsistent; sample sizes not given; *difference* (and prob. other rows) should report std\ error.
:::
`r if( !includeEvenAnswers) "-->"`

::::::




```{r TypingPlot, eval=includeGraphsTables, fig.cap="Plots for the typing dataset", fig.align="center", out.width='100%', fig.width=8, fig.height=4}
data(Typing)

par( mfrow = c(2, 3))

###

hist( Typing$mTS,
      xlab = "Typing speed (wpm)",
      ylab = "Number",
      main = "Typing speed",
      las = 1)

###

hist( Typing$mAcc * 100,
      xlab = "Typing accuracy (%)",
      ylab = "Number",
      main = "Typing accuracy",
      las = 1)

###

hist( Typing$Age,
      xlab = "Age of participants (years)",
      ylab = "Number",
      main = "Ages of participants",
      las = 1)

###

barplot( table(Typing$Sex),
         xlab = "Sex",
         ylab = "Number",
         main = "Sex of participants",
         names.arg = c("Male", "Female"),
         las = 1)

###

plot( Typing$mTS ~ Typing$Sex,
      xlab = "Sex",
      ylab = "Typing speed (wpm)",
      main = "Typing speed by sex",
      names = c("Male", "Female"),
      las = 1)

###

plot( (Typing$mAcc * 100) ~ Typing$Age,
      ylab = "Typing accuracy (%)",
      xlab = "Age",
      main = "Typing accuracy by age",
      las = 1)

```

```{r MinersGoodCrabs, eval=includeGraphsTables, fig.cap="Left: the number of noisy miners and the number of eucalyptus trees. Right: the colour of female horseshoe crabs and the condition of their spines. There are no missing values.", fig.align="center", out.width='100%', fig.width=9.25, fig.height=3}

par(mfrow = c(1, 2))

###
data(NMiner)

plot(Minerab ~ Eucs,
     data = NMiner,
     ylim = c(0, 20),
     las = 1,
     xlab = "No. of eucalypt trees",
     ylab = "No. of noisy miners",
     main = "No. eucalypts and the no. noisy miners in\n 2 ha. buloke woodland patches in Vic.",
     pch = 19)

###
data(HCrabs)

CrabsMat <- xtabs(~ Spine + Col,
                  data = HCrabs)

out <- barplot( CrabsMat,
                beside = FALSE,
                xlab = "Condition of the spine",
                ylab = "Carapace colour",
                main = "Carapace colour\nand spine condition",
                ylim = c(0, 170),
                axes = FALSE,
                col = grey( seq(0, 1, length = 3)),
                las = 1)
axis(side = 1,
     at = out,
     labels = colnames(CrabsMat) )
axis(side = 2,
     las = 1,
     at = seq(0, 100, by = 25))
legend("topleft",
       bty = "n",
       ncol = 3,
       cex = 0.9,
       fill = grey( seq(0, 1, length = 3)),
       legend = rownames(CrabsMat))
```



### Chap.\ \@ref(Probability): Probability {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ProbabilityMethod).**
**1.** Subjective. 
**2.** Rel.\ frequency.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ProbabilityMethodB).**
**1.** Classical.
**2.** Probably subjective.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ProbabilityAndOrNot).**
**1.** Just **Kings** and **Aces**.
**2.** $8/52 = 2/13$.
**3.** Picture cards.
**4.** $16/52 = 4/13$.
**5.** **Ace**, **King**, **Queen**, **Jack** of $\spadesuit$.
**6.** $4/52 = 1/13$.
**7.** Any $\heartsuit$, $\diamondsuit$ or $\clubsuit$.
**8.** $39/52 = 3/4$.
**9.** $4/16 = 1/4$.
**10.** $4/13$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ProbabilityAndOrNot2).**
**1.** No events.
**2.** $0$.
**3.** All $6$ events.
**4.** $1$.
**5.** 'Roll $2$'.
**6.** $1/6$.
**7.** 'Roll $1$, $3$, $4$, $5$, or $6$'.
**8.** $5/6$.
**9.** $1/3$.
**10.** $0$.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ProbabilityThreeEvents).**
**1.** Yes.
**2.** Yes.
**3.** $1/2$.
**4.** $1/2$.
**5.** HH, HT, TH, TT (Coin\ A listed first).
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ProbabilityThreeEvents2).**
**1.** $4/52 = 1/13$.
**2.** $1/13$.
**3.** $3/39 = 1/13$.
**4.** Independent.
**5.** $13/52 = 1/4$.
**6.** $0$.
**7.** $13/39 = 1/3$.
**8.** Not independent.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ProbabilityDie).**
**1.** $4/6$.
**2.** $5$. 
**3.** Yes: die outcome won't change coin outcome.
**4.** $1/2$.
**5.** $1/6$.
**6.** $1/3$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ProbabilityCards).**
$4/52 = 0.07692$.
$4/48 = 0.08333$.
$16/52 = 0.3077$.
$16/(52 - 16) = 0.4444$.
*Not* independent (like Ex.\ \@ref(exm:IndependenceCards)).
*Are* independent; what happens on the die does not change what happens with the cards.
$4/16 = 0.25$.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSpaceCardsDiff).**
**1.** In order drawn: BB, BR, RB, RR.
**2.** Equally-likely outcomes, so $1/2$.
**3.** $1/2$.
**4.** Yes.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSpaceInfiniteCoins).**
**1.** H, TH, TTH, TTTH, and so on.
**2.** $1/2$.
**3.** $1/4$.
**4.** $3/4$.
:::
`r if( !includeEvenAnswers) "-->"`
 
 
```{r echo=FALSE, results = 'hide'}
Tab <- array(dim = c(4, 2))
Tab[, 1] <- c(2540, 2734, 391, 362)
Tab[, 2] <- c(21219, 22574, 9496, 9963)

# Part 1
All <- sum(Tab)
FN <- sum( Tab[, 1])
notFN <- sum( Tab[, 2])

# Part 2
Gov <- sum( Tab[1:2, ])
nonGov <- sum( Tab[3:4, ])
Gov/All

# Part 3
Tab[1, 1] / sum(Tab[1, ])
Tab[2, 1] / sum(Tab[2, ])

# Part 4
Tab[3, 1] / sum(Tab[3, ])
Tab[4, 1] / sum(Tab[4, ])

# Part 5
Tab[1, 1]  / sum( Tab[ 1, ])
Tab[3, 1]  / sum( Tab[ 3, ])

# Part 6
Tab[2, 1]  / sum( Tab[ 2, ])
Tab[4, 1]  / sum( Tab[ 4, ])
```
 
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:FirstNationStudents).**
**1.** $0.087$.
**2.** $0.708$.
**3.** F: prob FN: $0.107$; M: prob FN: $0.108$; close to independent.
**4.** F: prob FN: $0.040$; M: prob FN: $0.035$; close to independent.
**5.** Gov: prob FN: $0.107$; NGov: prob FN: $0.040$; not independent.
**6.** Gov: prob FN: $0.108$; NGov: prob FN: $0.035$; not independent.
**7.** Regardless of sex, First Nations children more likely to be at government school.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoWayTableProbs).**
**1.** $49.2$%. 
**2.** $17.7$%. 
**3.** $55.1$%.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:IndependentEvents).**
**1.** *Not independent*: less likely to walk in rain.
**2.** *Not independent*: smoker far more likely to suffer from lung cancer than non-smoker.
**3.** *Not independent*: if it rains, I won't water garden.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SensitivitySpecifity).**
**1.** Expect $100\times 0.99 = 99$ people to return positive result.
**2.** Expect $900\times (1 - 0.98) = 18$ people to return positive result.
**3.** $18 + 99 = 117$ positive results.
Positive test result may, may not mean person has disease.
**4.** $99/117$, or $84$% of having disease.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CoinOutcomes).**
Reasoning assumes three *equally likely* outcomes (HH, TT, HT); untrue.
Consider tossing $20$-c coin (lower-case, normal) and  $1$-coin (capitals, **bold**).
*Four* outcomes:
h**H**,
h**T**,
t**H**
t**T**.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:AndrewsQuote).**
Events not equally likely.
:::
`r if( !includeEvenAnswers) "-->"`


::::::



### Chap.\ \@ref(SamplingVariation): Sampling variation {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:StdErrorOrStdDeviationA).**
**1.** Std\ dev.
**2.** Std\ error (of mean).
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:StdErrorOrStdDeviationB).**
**1.** Std\ dev.
**2.** Std\ error (of proportion).
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:HasStandardErrorA).**
**1.** No.
**2.** Yes.
**3.** Yes.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:HasStandardErrorB).**
**1.** Yes.
**2.** No.
**3.** Yes.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RouletteWheelA).**
**1.** Reasonable, if fair.
**2.** Almost impossible, if fair.
**3.** Unlikely (not impossible), if fair.
**4.** Highly unlikely, if fair.
::::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RouletteWheelB).**
**1.** Reasonable, if fair.
**2.** Highly unlikely, if fair.
**3.** Reasonable, if fair.
**4.** Highly unlikely, if fair.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:QuoteStdError).**
*Std\ error of the mean* describes how sample mean varies from sample to sample.
:::

::::::



<!-- \pagebreak MANUAL LATEX PAGE BREAK ADDED --> 

### Chap.\ \@ref(SamplingDistributions): Models and normal distributions {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Statements).**
Only **1.** and **2.** false.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:StatementsB).**
Only **1.** and **3.** false.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:BasiczA).** 
**1.** $0.9671$.
**2.** $0.0183$.
**3.** Close to zero.
**4.** Close to one.
:::
<!-- round(pnorm( c(1.84, -2.09, -5.34, 4.25) ), 4) -->


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:BasiczA).** 
**1.** $0.6844$.
**2.** $0.1515$.
**3.** $0.99997$, using note at bottom of table.
**4.** $0.5$.
:::
`r if( !includeEvenAnswers) "-->"`
<!-- 1 - round(pnorm( c(-0.48, 1.03, -4, 0) ), 4) -->


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsGrowthChartA).**
About $2.5$% of girls under $100\cms$ tall.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsGrowthChartB).**
About $2.5$% of girls taller than $120\cms$.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsIQForwards).**
**1:** C; **2:** A; **3:** B; **4:** D.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsIQBackwards).**
**1:** A; **2:** C; **3:** B; **4:** D.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsEmpiricalA).**
$68.26$%; very close to $68$%.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsEmpiricalB).**
$95.44$%;; very close to $95$%.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsTrees).**
**1.** $z = -0.30$; about $38.2$%.
**2.** $z = 0.07$; about $47.2$%.
**3.** $z = -0.67$ and $z = 0.44$; about $41.9$%.
**4.** About $z = 1.04$; diameter about $11.6$\ inches.
**5.** About $z = -0.67$; diameter about $7.0$\ inches.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CornSeeds).**
**1.** $3.1$%.
**2.** $3.8$%.
**3.** $48.7$%.
**4.** Smaller than  $7.38\mms$.
**5.** Larger than $7.21\mms$.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsGestationLength).**
**1.** $z = -0.61$; $72.9$%.
**2.** $z = -1.83$; $3.4$%.
**3.** $z = -4.878$ and $z = -1.83$; $3.4$%.
**4.** $z = 1.64$ (or $1.65$); $5$% *longer* than $42.7$ weeks. 
**5.** $z$-score: $-1.28$; $10$% *shorter* than $37.9$ weeks.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsBridgesTrucks).**
$-1.77$; $10$; mass lower than average of $14$ tonnes.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsIQs).**
$z = 2.05$.
IQ: $130.75$.
$\text{IQ} > 130$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsIQsMilitary).** 
Lower than $80.8$: rejection.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionsChargingEVs).**
Use *number minutes from (say) 5:30pm*.
Std\ dev.: $120\mins$, plus $0.28\times 60 = 16.8\mins$ = $136.8\mins$.
**1.** $9$pm; $210$mins$ from $5$:$30$pm; $z = 1.54$; $6.2$%.
**2.** $z = -0.22$; $41.3$%.
**3.** $z_1 = -0.22$ and $z_2 = 0.22$; $0.5871 - 0.4129$; $17.4$%.
**4.** $z$-score: $0.52$; $x  = 71.136$ mins after $5$pm; about one hour and $11$\ mins after $5$:$30$pm, or $6$:$41$pm.
**5.** $z$-score: $-1.04$; $x = -141.272$, or $141.272\mins$ *before* $5$:$30$pm; about two hours and $21$\mins before $5$:$30$pm, or $3$:$09$pm.
:::

::::::





### Chap.\ \@ref(CIOneProportion): Confidence intervals: one proportion {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIOneProportionHiccups).**
$\hat{p} = 0.81944$, $n = 864$.
$\text{s.e.}(\hat{p}) = 0.01309$; approx. $95$%\ CI: $0.819 \pm (2\times 0.0131)$ or $0.793$ to\ $0.845$.
Stat.\ valid.
(Many decimal places used for working; final answers rounded.)
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIParamedic).**
$\hat{p} = 791/1766 = 0.44790$, $n = 1766$, $\text{s.e.}(\hat{p}) =  0.011833$.
*Approx.* $95$%\ CI $0.424$ to\ $0.472$.
(Many decimal places used for working; final answers rounded.)
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIzA).**
$z = 1.96$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIzB).**
$z = 2.57$ or $2.58$.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIOneProportionSnacking).**
$\hat{p} = 0.051948$; $\text{s.e.}(\hat{p}) = 0.00178$; approx. $95$%\ CI: $0.0519\pm 0.0358$.
Stat.\ valid.
:::
```{r eval = FALSE}
phat <- 8/154; se <- sqrt( (phat *(1 - phat))/ 154)
phat; se;
phat + c(-2, 2) * se
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:KoalasCrossingRoads).**
$\hat{p} = 18/51 = 0.3529$; $\text{s.e.}(\hat{p}) = 0.06692$.
Approx. $95$%\ CI: $0.3529 \pm 0.1338$.
*Margin of error*: $0.1338$.
Approx. $95$%\ CI $0.219$ to\ $0.487$.
Stat.\ valid.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval = FALSE}
phat <- 18/51; se <- sqrt( (phat *(1 - phat))/ 51)
phat; se;
phat + c(-2, 2) * se
```



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIOneProportionSaltIntake).**
$\hat{p} = 0.317059$; $n = 6882$.
$\text{s.e.}(\hat{p}) =  0.0056092$.
CI: $0.317\pm 0.011$.
Stat.\ valid.
:::
```{r eval = FALSE}
phat <- 2182/6882; se <- sqrt( (phat *(1 - phat))/ 6882)
phat; se;
phat + c(-2, 2) * se
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CITurbines).**
$3\,000\hs$: $\hat{p} = 0.2143$; $\text{s.e.}(\hat{p}) = 0.06331$.
CI: $0.088$ to\ $0.341$.
Stat.\ valid.
$400\hs$: $\hat{p} = 0$; $\text{s.e.}(\hat{p}) = 0$.
CI: $0$ to\ $0$: silly (no sampling variation?): *not* stat.\ validity conditions.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval = FALSE}
phat <- 9/42; se <- sqrt( (phat *(1 - phat))/ 42)
phat; se;
phat + c(-2, 2) * se

phat <- 0/39; se <- sqrt( (phat *(1 - phat))/ 39)
phat; se;
phat + c(-2, 2) * se
```



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CanadianEnergyDrinks).**
$\hat{p} = 0.241$.
$\text{s.e.}(\hat{p}) =  0.010984$.
Approx. $95$%\ CI: $0.219$ to\ $0.263$.
:::
```{r eval = FALSE}
phat <- 365/1516; se <- sqrt( (phat *(1 - phat))/ 1516)
phat; se;
phat + c(-2, 2) * se
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OnePropCIAlcohol).**
$\hat{p} = 0.49310$.
$\text{s.e.}(\hat{p}) =  0.015700$.
Approx. $95$%\ CI: $0.462$ to\ $0.524$.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval = FALSE}
phat <- 500/1014; se <- sqrt( (phat *(1 - phat))/ 1014)
phat; se;
phat + c(-2, 2) * se
```


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OnePropCIAI).**
$\hat{p} = 0.3182$.
$\text{s.e.}(\hat{p}) =  0.0702175$.
Approx. $95$%\ CI: $0.178$ to\ $0.459$.
:::
```{r eval = FALSE}
phat <- 14/44; se <- sqrt( (phat *(1 - phat))/ 44)
phat; se;
phat + c(-2, 2) * se
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OnePropCICheating).**
$\hat{p} = 0.064356$.
$\text{s.e.}(\hat{p}) =  0.017265$.
Approx. $95$%\ CI: $0.0298$ to\ $0.0989$.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval = FALSE}
phat <- 13/202; se <- sqrt( (phat *(1 - phat))/ 202)
phat; se;
phat + c(-2, 2) * se
```

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OnePropCIWearHats).**
$\hat{p} = 0.13431$.
$\text{s.e.}(\hat{p}) = 0.012434$.
Approx. $95$%\ CI: $0.109$ to\ $0.159$.
:::
```{r eval = FALSE}
phat <- 101/752; se <- sqrt( (phat *(1 - phat))/ 752)
phat; se;
phat + c(-2, 2) * se
```

`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OnePropCIWearSunglasses).**
$\hat{p} = 0.33112$.
$\text{s.e.}(\hat{p}) =  0.017162$.
Approx. $95$%\ CI: $0.297$ to\ $0.365$.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval = FALSE}
phat <- 249/752; se <- sqrt( (phat *(1 - phat))/ 752)
phat; se;
phat + c(-2, 2) * se
```

::::::



```{r, eval=includeGraphsTables, fig.width=4, fig.height=2.5, out.width='50%', fig.align="center", fig.cap="Proportion of males"}
z <- seq(-3.5, 3.5,
         length = 100)
mn <-  0.8194444
sdev <- 0.01308604
y <- dnorm(z)

z.special <- seq(-3, 3,
                 by = 1)
labs <- round(mn + (z.special * sdev), 3)

plot( y ~ z,
      axes = FALSE,
      type = "l",
      lwd = 2,
      ylab = "",
      xlab = "Proportion of males")
axis(side = 1,
     at = seq(-3, 3, by = 1),
     cex = 0.9,
     labels = labs)
abline(v = 0,
       lwd = 2,
       col = "grey")
abline(v = c(-1, 1),
       lwd = 1,
       col = "grey")
abline(v = c(-2, 2),
       lwd = 1,
       col = "grey")
abline(v = c(-3, 3),
       lwd = 1,
       col = "grey")
```





### Chap.\ \@ref(OneMeanConfInterval): Confidence intervals: one mean {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneMeanCIBears).**
**1.** *Parameter*: pop. mean weight of American black bear, $\mu$.
**2.** $\text{s.e.}(\bar{x}) = 3.756947$.
**3.** Normal; mean $\mu$; std dev: $3.757$,
**4.** $77.4$ to\ $92.4\kgs$.
**5.** Approx. $95$% confident *population mean* weight of male American black bears between $77.4$ and $92.4\kgs$.
**6.** Stat.\ valid: $n \ge 25$.
:::
```{r eval = FALSE}
xbar <- 84.9; n <- 185; s <- 51.1
se <- s/sqrt(n); se
xbar + c(-2, 2) * se
```



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:BagsCI).**
**1.** Pop. mean weight of school bags for Iranian children Grades $6$--$8$.
**2.** $\text{s.e.}(\bar{x}) = 0.03883\kgs$.
**3.** Normal; mean $\mu$; std dev: $0.03883\kgs$.
**4.** CI: $2.8\pm0.07766\kgs$.
**5.** $\bar{x} = 2.8\kgs$ (s.e.: $0.0388$; $n = 586$); approx. $95$%\ CI $2.72$ to\ $2.88\kgs$.
**6.** Stat.\ valid: $n \ge 25$.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval = FALSE}
xbar <- 2.8; n <- 586; s <- 0.94
se <- s/sqrt(n); se
xbar + c(-2, 2) * se
```


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIOneMeanLungCapacityInChildren).**
$\text{s.e.} = 0.06410$.
*Approx.* $95$%\ CI: $2.72\Ls$ to\ $2.98\Ls$.
:::
```{r eval = FALSE}
xbar <- 2.85; n <- 45; s <- 0.43
se <- s/sqrt(n); se
xbar + c(-2, 2) * se
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIOneMeanEnvironmentalPollution).**
Std error: $\text{s.e.} = 994.2182$ (use lots of decimal places in working).
*Approx.* $95$%\ CI: $4\,967.984$\c$\mu$g to\ $8\,944.86$\ $\mu$g.
Not likely to apply elsewhere; park near lead smelter.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval = FALSE}
xbar <- 6956.41; n <- 58; s <- 7571.74
se <- s/sqrt(n); se
xbar + c(-2, 2) * se
```


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIOneMeanToothbrushing).**
Approx. $95$%\ CI: $29.9$ to\ $36.1\secs$.
:::
```{r eval = FALSE}
xbar <- 33.0; n <- 60; s <- 12.0
se <- s/sqrt(n); se
xbar + c(-2, 2) * se
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIOneMeanBloodLoss).**
**1.** Approx. $95$%\ CI: $754.1\mLs$ to\ $938.7\mLs$.\spacex
**2.** Not good at estimating (from article: guesses between $50$ to $3\,000\mLs$).\spacex
**3.** Stat.\ valid.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval = FALSE}
xbar <- 846.4; n <- 199; s <- 651.1
se <- s/sqrt(n); se
xbar + c(-2, 2) * se
```


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneMeanCINHANESInterpret).**
*None* acceptable.
**1.** CIs not about observations, but *statistics*.
**2.** CIs not about observations, but *statistics*.
**3.** *Samples* don't vary between values; *statistics* do.
(CIs about populations anyway.)
**4.** *Populations* don't vary between values. 
**5.** Parameters do not vary. 
**6.** *Know* $\bar{x} = 1.3649\mmols$/L.\spacex
**7.** *Know* $\bar{x} = 1.3649\mmols$/L.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneMeanStdError).**
*Neither* correct.
For variation in *individuals* trees, use *std deviation* not std error.
CIs about pop. mean diameter, not individual trees.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ChewingTime).**
$\text{s.e.}(\bar{x}) = 5.36768$; approx. $95$%\ CI: $50.56$ to\ $72.04\secs$.
Stat.\ valid.
:::
```{r eval = FALSE}
xbar <- 61.3; n <- 30; s <- 29.4
se <- s/sqrt(n); se
xbar + c(-2, 2) * se
```



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIPizzas).**
**1.** $\mu$: mean diameter of *all* EB pizzas; 
$\bar{x}$: mean diameter of sample pizzas.
**2.** $\bar{x} = 11.486$ inches (to $0.001$ of a cm?).
       Value $\mu$ unknown; claimed $\mu = 12$.
**3.** $s = 0.2465$ inches; $\sigma$ unknown value.
**4.** $0.02205$.
**5.** $s$: variation in diameters of individual pizzas;
       s.e.: precision of sample mean to estimate pop. mean.
**6.** Almost certainly not the same; probably close to $11.486$ inches.
**7.** Normal; mean $\mu$; std\ dev.: std error $0.02205$.
**8.** Approx. $95$%\ CI: $11.44$ to\ $11.53$ inches.
**9.** Not given.
**10.** $n \ge 25$, or pop. has normal distribution; stat.\ valid.
**11.** $\mu$ probably not $12$ inches.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIMatchesPerBox).**
**1.** *One* observation $x = 44$; claimed *population mean* is $\mu = 45$.
**2.** OK to have decimal value as *mean*.
**3.** $\bar{x} = 44.9$; $\mu = 45$: different things; why should they be same?
**4.** CI allows for sampling variation.
**5.** $44.850$ to\ $44.950$.
**6.** Possibly lying; not certain.
**7.** Sample mean; $\bar{x} = 44.9$.
**8.** Population mean; value unknown, but *claimed* to be $\mu = 45$.
:::

::::::








### Chap.\ \@ref(AboutCIs): More details about CIs {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:AboutCIsInterpretationP).**
**1.** CIs are intervals for unknown *parameters*, not known *statistics*. 
**2.** CIs for proportion (or percentage), not *number* of trees.
       (The CI is $68$% anyway, not $95$%.)
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:AboutCIsInterpretationP2).**
Sample proportion *is* $0.981$; need CI for parameter $p$.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:AboutCIsInterpretationMean).**
**1.** CIs *not* about individuals.
**2.** CIs *not* about *sample* means.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:AboutCIsInterpretationMean2).**
**1.** CIs *not* about individuals.
**2.** CIs *not* about *sample* means.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CIPossums).**
Intervals for different things.
First: $95$%\ CI for *mean* weight.
Second: *not* CI; for weights of *individuals* possums.
:::

::::::







\pagebreak <!--               MANUAL           -->







### Chap.\ \@ref(MakingDecisions): Making decisions {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MakingDecisionsDice)**.
**1.** Yes; very likely (can't be sure).
**2.** Assuming fair die, *not* expect `r if (knitr::is_latex_output()) {
   '\\largedice{6}'
} else {
   '<span class="larger-die">&#9861;</span>'
}` ten times in row.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MakingDecisionsCoin)**.
**1.** No; seem reasonable (can't be sure).
**2.** Assuming fair coin, two consecutive heads is not unusual.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MakingDecisionsJuryService)**.
Seems unlikely.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MakingDecisionsClaim)**.
**1.** Mean diameter $12$\ inches (as claimed).
**2.** Difference due to sampling variation; difference because mean not really $12$\ inches.
**3.** Perhaps unsupported?
**4.** Very likely unsupported.
**5.** Yes.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MakingDecisionsOlder)**.
**1.** $p = 0.36$.
**2.** $\hat{p} = 0.433$; probably not.
**3.** $\hat{p} = 0.1$; probably.
:::
::::::






### Chap.\ \@ref(TestOneProportion): Hypothesis tests: one proportion {.unlisted .unnumbered}


:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:sepPWhy1).**
In tests, $p$ assumed known.
In CI, have no value for $p$ to use.
:::


`r if( !includeEvenAnswers) "<!--"`
:::{.exercise #sepPWhy2}
No value of $p$ to use
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExplainA).**
$0.38$ is *sample* proportion; RQ asks about *pop.* proportion of $1/6$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExplainB).**
$p$ has no standard error; does not vary.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesDodgyA).**
Tests *not* about sample value (we *know* value of $\hat{p}$), but about unknown pop. value (i.e., $p$).
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesDodgyB).**
'The null hypothesis equal to $0.076$' makes no sense; mean: 'the value of some *parameter* is equal to $0.076$'.
Also, $0.076$ appears to be sample value, but hypotheses should be formed before data collected.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesPlacebos).**
**1.** One-in-five: $0.2$.
**2.** $H_0$: $p = 0.2$; $H_1$: $p > 0.2$.
**3.** One-tailed.
**4.** Normal; mean $0.2$, std\ deviation $\text{s.e.}(\hat{p}) = 0.0444$.
**5.** $\hat{p} = 0.6173$; $z = 9.39$: $P$ *very small*:
Very strong evidence people do better-than-guessing at identifying placebo.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:POnePropTestMeasles).**
**1.** $\hat{p} = 55/972 = 0.056584...$.
**2.** $H_0$: $p = 0.10$ (i.e, assume target met; diff between $p$ and $\hat{p}$ due to sampling variation); $H_1$: $p < 0.10$.
       One-tailed: RQ is whether target is $10$% or *lower*.
**3.** $\text{s.e.}(\hat{p}) = 0.0096225...$.
**4.** $z = -4.51$; *very* large (and *negative*) $z$-score; $P$ very small.
**5.** Very strong evidence in sample ($z = -4.51$; one-tailed $P < 0.001$) pop.\ proportion less than target of $p = 0.10$ (Korean sample proportion: $\hat{p} = 0.0566$; $n = 972$; approx. $95$%\ CI $0.042$ to $0.071$).
**6.** Stat.\ validity: $n\times p = 97.2$ and $n\times (1 - p) = 874.8$; both exceed $5$.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestTurtleSex).**
$H_0$: $p = 0.5$; $H_1$: $p \ne 0.5$.
$\hat{p} = 0.39726$; $\text{s.e.}(\hat{p}) = 0.05727$; $z = -1.794$.
$P$ not that small.
No evidence of difference.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesEPL).**
$H_0$: $p = 0.5$; $H_1$: $p > 0.5$ (*one*-tailed).
$\hat{p} = 0.5759$; $n = 158$.
$\text{s.e.}(\hat{p}) = 0.03977786$, so $z = 1.91$.
Using tables, one-tailed $P = 0.0281$; slight evidence of home-ground advantage in pop.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesPedalMachines).**
$H_0$: $p = 0.0602$; $H_1$: $p < 0.602$ (*one*-tailed).
$\hat{p} = 0.5008489$; $n = 589$: $\text{s.e.}(\hat{p}) = 0.0201689$, so $z = -5.015$.
$P$ very small.
Strong evidence prop. F using machines lower than prop. F in uni pop.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesCasinos).**
$H_0$: $p = 0.255$; $H_1$: $p \ne 0.255$.
$\hat{p} = 0.2464986$; $n = 357$, so $\text{s.e.}(\hat{p}) = 0.0230682$, so $z = -0.368533$.
Very small; $P$ large.
No evidence proportion of smokers different for casino-goers compared to general US population.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionBreadfruitPasta).**
$H_0$: $p = 0.5$; $H_1$: $p > 0.5$ (one-tailed).
$\hat{p} = 0.802817$; $n = 71$: $\text{s.e.}(\hat{p}) = 0.0593391$, so $z = 5.10$.
$P$ very small.
Strong evidence majority like breadfruit pasta (for pop. represented by sample anyway).
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesCTS).**
$H_0$: $p = 0.15$ and $H_1$: $p \ne 0.15$.
$\hat{p} = 0.0639535$ and $n = 516$: $\text{s.e.}(\hat{p}) = 0.0157192$, so $z = -5.473$.
$P$ very small.
Strong evidence prop. people with CTS with a PL tendon absent is different for people with CTS.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesBorers).**
$H_0$: $p = 1/16 = 0.625$; $H_1$: $p \ne 0.0625$.
$\hat{p} = 0.139535$ and $\text{s.e.}(\hat{p}) = 0.018457$; $z = 26.3$: *massive*; $P$ very small.
Very strong evidence pop.\ proportion not $1/16$; borers *not* resistant.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesLEDlights).**
$H_0$: $p = 0.5$; $H_1$: $p \ne 0.5$.
$\hat{p} = 0.557692$; $\text{s.e.}(\hat{p}) = 0.0693375$, giving $z = -0.832$.
$P$ 'large'.
No evidence to suggest choice non-random.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesCoinSpin).**
$\hat{p} = 0.56$; $z = 1.91$.
Slight evidence of bias.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneProportionTestExercisesBirths).**
**1.** $105.5/100 = 0.51338$.
**2.** $\hat{p} =   0.51328$.
**3.** $z = -0.1084$; no evidence of any variation from long-term proportion.
:::
`r if( !includeEvenAnswers) "-->"`

::::::



### Chap.\ \@ref(TestOneMean): Hypothesis tests: one mean {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneTSpeed).**
**1.** $\mu$, pop. mean speed (km.h^$-1$^).
**2.** $H_0$: $\mu = 90$; $H_1$: $\mu > 90$ (one-tailed).
**3.** $\text{s.e.}(\bar{x}) = 0.6937$.
**5.** $t = 9.46$.
**6.** $t$-score *huge*; (one-tailed) $P$ very small: very strong evidence mean speed of vehicles on road greater than $90\kms.\xhs^{-1}$.
**7.** Stat.\ valid.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestOneMeanExercisesSlalom).**
$H_0$: $\mu = 38.8$; $H_1$: $\mu \ne 38.8$ (two-tailed), where $\mu$ mean distance in metres.
Std error: $\text{s.e.}(\bar{x}) = 0.37792$.
$t$-score: $-5.98$, *huge*; $P$ *very* small.

*Very* strong evidence ($P < 0.001$) that GPS is not recording distance correctly.
Since $n > 25$, stat.\ valid.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestOneMeanExercisesAutomatedVehicles).**
$H_0$: $\mu = 50$; $H_1$: $\mu > 50$ (one-tailed).
$\text{s.e.}(\bar{x}) = 4.701076$. 
$t = 7.23$: $P$ very small.
*Very* strong evidence ($P < 0.001$) mean mental demand *greater* than $50$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestOneMeanWaterTemp).**
$t = -8.5$; very small $P$.
Evidence mean water temp below $60$^o^C.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestOneMeanExercisesCherryRipes).**
$H_0$: $\mu = 14$; $H_1$: $\mu \ne 14$ (two-tailed).
$\text{s.e.}(\bar{x}) = 0.092493$.
$t$-score: $10.35$: *huge*; $P$ very small.
*Very* strong evidence ($P < 0.001$) mean weight of Fun Size *Cherry Ripe* bar not $14\gs$.
SD: the variation in weight of individual bars.
SE: the variation in sample means for $n = 67$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestOneMeanBloodLoss).**
$H_0$: $\mu = 1\,000$ and $H_1$: $\mu \ne 1\,000$ ($\mu$: pop.\ mean guess of spill volume).
Std\ error: $46.15526$.
$t = -3.33$: $P$ very small.
Very strong evidence mean guess of blood volume not $1\,000\mLs$, the actual value.
Stat.\ valid.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestOneMeanExercisesSleep).**
$H_0$: $\mu = 10$ (or $\mu \ge 10$) and $H_1$: $\mu < 10$.
*F*: $\text{s.e.}(\bar{x}) = 0.05924742$; $t = -25.32$.
*M*: $\text{s.e.}(\bar{x}) = 0.0700152$;  $t = -19.42$. 
Both $P$ extremely small.
For both boys and girls, very strong evidence mean sleep time on weekend less than $10\hs$.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestOneMeanQualityControl).**
Hypotheses have form $H_0$: $\mu = \text{pre-determined target}$, $H_1$: $\mu \ne \text{pre-determined target}$.
$t$-scores: $t_1 = 0.318$, $t_2 = 2.347$, $t_3 = -0.466$, $t_4 = -0.726$.
$P$ large, except for $2$nd test.
No evidence instruments are dodgy, except perhaps for $1$st instrument, mid-level LH concentrations.
Should be stat.\ valid.
Assessing means is useful, how *variable* the measurements also useful (but beyond us).
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:PizzasHT).**
**1.** $\mu$:  population mean pizza diameter.
**2.** $\bar{x} = 11.486$; $s = 0.247$.
**3.** $0.02205479$.
**4.**  $H_0$: $\mu = 12$; $H_1$: $\mu\ne 12$.
**5.** Two-tailed; RQ asks if diameter is $12$ inches, or not.
**6.** Normal distribution, mean $12$ and std dev.\ of $\text{s.e.}(\bar{x}) = 0.02205$.
**7.** $t = -23.3$.
**8.** $P$ *really* small.
**9.** Very strong evidence mean diameter not $12$\ inches.
**10.** $n$ much larger than $25$; stat.\ valid.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OneMeanHtExerciseSleepTime).**
School days: $t = -55.78$.
*Very* strong evidence guidelines not met.
Non-school days: $t = 24.44$ (sample mean *greater than* recommendation).
No evidence guidelines *not* met.
:::
`r if( !includeEvenAnswers) "-->"`


::::::

```{r eval=includeGraphsTables}
data(LHconc)

DataAndTargets <- array(dim = c(3, 4))
DataAndTargets[1, ]<- c(64.31,
                        19.24,
                        64.97,
                        19.40)
DataAndTargets[2, ]<- c(1.700,
                        0.588,
                        1.029,
                        0.413)
DataAndTargets[3, ]<- c(64.22,
                        19.01,
                        65.05,
                        19.45)

n <- length(LHconc$High1)

rownames(DataAndTargets) <- c("Mean of data",
                              "Std dev. of data",
                              "Pre-determined target")
colnames(DataAndTargets) <- c("High (Instrument 1)",
                              "Mid (Instrument 1)",
                              "High (Instrument 2)",
                              "Mid (Instrument 2)")

ts <- ( DataAndTargets[1,] - DataAndTargets[3,] ) / ( DataAndTargets[2,]/sqrt(n))
```




### Chap.\ \@ref(MoreAboutTests): More details about hypothesis tests {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutExercisesApproximatingPValues).**
Use $68$--$95$--$99.7$ rule and diagram:
**1.** Very small; certainly less than $0.003$ ($99.7$% between $-3$ and\ $3$).
**2.** Very small; bit bigger than $0.003$ ($99.7$% between $-3$ and\ $3$).
**3.** Bit smaller than $0.05$ ($95$% between $-2$ and\ $2$).
**4.** *Very* small; *much* smaller than $0.003$.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutExercisesApproximatingPValues2).**
Using the $68$--$95$--$99.7$ rule and diagram:
**1.** Bit smaller than $0.32$ ($68$% between $-1$ and\ $1$).
**2.** Large-ish: between $0.32$ ($68$% between $1$ and\ $-1$) and $0.05$% ($95$% between $-2$ and\ $2$), but closer to $0.32$.
**3.** *Very* small; *much* smaller than $0.003$.
**4.** *Very* large! Almost $0.50$.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutExercisesApproximatingPValuesOneTailed).**
*Half* values in Ex.\ \@ref(exr:MoreAboutExercisesApproximatingPValues).
**1.** Very small; certainly less than $0.0015$.
**2.** Very small; bit bigger than $0.0015$.
**3.** Bit smaller than $0.025$.
**4.** *Very* small; *much* smaller than $0.0015$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutExercisesApproximatingPValuesOneTailed2).**
*Half* values in Ex.\ \@ref(exr:MoreAboutExercisesApproximatingPValues2).
**1.** Bit smaller than $0.16$.
**2.** Between $0.16$ and $0.025$; closer to $0.32$.
**3.** *Very* small; *much* smaller than $0.0015$.
**4.** *Very* large! Almost $0.25$.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutTestsInterpretingResults).**
$P$ *just larger* than $0.05$; 'slight evidence' to support $H_1$.
$P$ *just smaller* than $0.05$; 'moderate evidence' to support $H_1$.
The difference between $0.0501$ and\ $0.0499$ trivial.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutTestsInterpretingResults2).**
$P$ *just larger* than $0.05$; 'moderate evidence' to support $H_1$.
$P$ *just smaller* than $0.05$; 'strong evidence' to support $H_1$.
Diff between $0.011$ and\ $0.009$ trivial.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutTestsInterpretingHypotheses).**
**1.** Hypotheses about *parameters*. 
**2.** RQ two-tailed.
**3.** $36.8052$ is sample mean; hypothesis written *before* data collected.
**4.** Hypotheses about parameters; $36.8052$ is sample mean.
       These test if *sample* mean is $36.8052$; we *know* it is.
**5.** Hypothesis written *before* data collected.
**6.** Hypotheses about parameters.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutTestsInterpretingHypotheses2).**
**1.** Hypotheses about *parameters* like $p$. 
**2.** No parameter to test.
**3.** Hypotheses about *parameters*.
**4.** Hypotheses about *parameters*
**5.** A two-tailed RQ.
**6.** $0.41$ computed from sample.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutTestsConclusions).**
**1.** Conclusion about pop.\ **mean** energy intake.
**2.** Conclusions *never* about statistics. 
**3.** Conclusion about pop.\ **mean** energy intake.
**4.** $P$ *is* $0.018$, not *less than* $0.018$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutTestsBatteries).**
**1.** *No* evidence of a diff in lifetime between two brands: $P$ 'large'.
**2.** No; diff is $0.29\hs$, or about $17\mins$.
**3.** A diff of $17\mins$ in over $5\hs$ of use: trivial.
**4.** Conclusion: no evidence of diff between mean lifetimes; cumbersome for advertising!
       Common trick: 'No other battery lasts longer!'.
**5.** Price!
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutTestsConsistency).**
Statements\ 1 and\ 3 consistent.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MoreAboutTestsConsistency2).**
Statements\ 1 and\ 2.
:::
`r if( !includeEvenAnswers) "-->"`

::::::




```{r BroccoliSummaryTable, eval=includeGraphsTables}
BroccoliSummaryTable <- array( NA,
                               dim = c(3,3))
rownames(BroccoliSummaryTable) <- c("Raw",
                                    "With dip",
                                    "Differences")
colnames(BroccoliSummaryTable) <- c("Mean",
                                    "Standard deviation",
                                    "Standard error")
BroccoliSummaryTable[1, ] <- c(56.0,
                               26.6,
                               round(26.6/sqrt(101), 2))
BroccoliSummaryTable[2, ] <- c(61.2,
                               28.7,
                               round(28.7/sqrt(101), 2))
BroccoliSummaryTable[3, ] <- c(5.2,
                               NA,
                               3.06)

tab.cap.br <- "A  numerical summary for the Broccoli data"

if( knitr::is_latex_output() ) {
  kable( pad(BroccoliSummaryTable,
             surroundMaths = TRUE,
             targetLength = c(4, 4, 4),
             decDigits = c(1, 1, 2)),
         format="latex",
         booktabs = TRUE,
         align = "c",
         digits = c(1, 1, 2),
         longtable = FALSE,
         escape = FALSE,
         col.names = colnames(BroccoliSummaryTable),
         caption = tab.cap.br) %>%
    column_spec(1, bold = TRUE)  %>%
    row_spec(0, bold = TRUE) %>%
    kable_styling(font_size = 8)
}
if( knitr::is_html_output() ) {
  kable( pad(BroccoliSummaryTable,
             surroundMaths = TRUE,
             targetLength = c(4, 4, 4),
             decDigits = c(1, 1, 2)),
                format = "html",
             align = "c",
                booktabs = TRUE,
                digits = c(1, 1, 2),
                longtable = FALSE,
                col.names = colnames(BroccoliSummaryTable),
                caption = tab.cap.br) %>%
    column_spec(1, bold = TRUE)
}
```



```{r FruitSummary, eval=includeGraphsTables}
data(Fruit)

Fruit$Farm <- 1 : length(Fruit$FWeight2014)

FruitTab <- dplyr::select(Fruit,
                          Farm,
                          FWeight2014,
                          FWeight2015)
FruitTab$Change <- FruitTab$FWeight2015 - FruitTab$FWeight2014

FruitSummary <- array( dim = c(3, 4))

rownames(FruitSummary) <- c("Dry year (2014)",
                            "Normal year (2015)",
                            "Change")
colnames(FruitSummary) <- c("Mean",
                            "Std dev.",
                            "Std error",
                            "Sample size")

FruitSummary[, 1] <- colMeans(FruitTab[, 2:4],
                              na.rm = TRUE)
FruitSummary[, 2] <- apply(FruitTab[, 2:4],
                           2,
                           "sd",
                           na.rm = TRUE)
FruitSummary[, 3] <- apply(FruitTab[, 2:4],
                           2,
                           "findStdError",
                           na.rm = TRUE)
FruitSummary[, 4] <- apply(FruitTab[, 2:4],
                           2,
                           "realLength")

# Do some appropriate rounding
FruitSummary <- round(FruitSummary, 4)
FruitSummary[1:2, 1] <- round(FruitSummary[1:2, 1], 3)

if( knitr::is_latex_output() ) {

  knitr::kable(pad(FruitSummary,
                   surroundMaths = TRUE,
                   targetLength = c(7, 6, 6, 2),
                   decDigits = c(3, 3, 3, 0)),
               format = "latex",
               align = "c",
               linesep = "",
               caption = "The fruit weight (in g) at farms for two different years",
               col.names = c("Mean", "Std dev.", "Std error", "Sample size"),
               row.names = TRUE,
               escape = FALSE,
               booktabs = TRUE) %>%
    row_spec(0, bold = TRUE) %>%
    row_spec(3, italic = TRUE) %>%
    kable_styling(font_size = 8)

}

if( knitr::is_html_output() ) {
  kable( pad(FruitSummary,
                   surroundMaths = TRUE,
                   targetLength = c(7, 6, 6, 2),
                   decDigits = c(3, 3, 3, 0)),
         format = "html",
         align = "c",
         booktabs = TRUE,
         longtable = FALSE,
         col.names =  c("Mean", "Std dev.", "Std error", "Sample size"),
         caption = "The fruit weight (in g) at farms for two different years") %>%
    row_spec(0, bold = TRUE)
}
```



```{r, FruitPlots, eval=includeGraphsTables, out.width='90%', fig.width=9, fig.height=3.5, fig.align="center", fig.cap="The fruit-weight data. Left: a histogram of the weight changes on each farm. Right: a case-profile plot of the weight changes where each line corresponds to a farm (unfilled points and dashed lines indicate a decrease)."}
par( mfrow = c(1, 2))

 hist(FruitTab$Change,
      las = 1,
      xlab = "Weight increase (in g)",
      ylab = "Number of farms",
      main = "Histogram of fruit weight\nincrease from 2014 to 2015")

 ###

 plot( x = c(0.75, 2.25),
       y = c(150, 400),
       type = "n",
       las = 1,
       xlab = "",
       main = "Case-profile plot of fruit weight\nincreases from 2014 to 2015",
       ylab = "Farm fruit weight (in g)",
       axes = FALSE)
 axis(side = 1,
      at = c(1, 2),
      labels = c("2014 (dry)",
                 "2015 (normal)"))
 axis(side = 2,
      las = 1)
 box()


 for (i in (1:length(FruitTab$Change))){
   if (!is.na(FruitTab$Change[i])){
     delta <- ifelse(FruitTab$Change[i] > 0, 0.025, -0.025)
     lines( x = c(1 + delta, 2 - delta),
            y = c( FruitTab$FWeight2014[i],
                   FruitTab$FWeight2015[i]),
            lty = ifelse(FruitTab$Change[i] > 0, 1, 2),
            pch = ifelse(FruitTab$Change[i] > 0, 19, 1),
            type = "b",
            lwd = 1)
   }
 }

```







\pagebreak <!--               MANUAL           -->







### Chap.\ \@ref(AnalysisPaired): CIs and tests: mean differences (paired data) {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeanDiffWhichPaired).**
**1.** Paired.
**2.** Paired.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeanDiffWhichPaired2).**
**1.** Not paired.
**2.** Paired.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeanDiffGDiffsA).**
How much longer task takes on the PC.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeanDiffGDiffsB).**
How much larger pH is downstream compared to upstream.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeanDiffFlowering).**
CI calculated as: $-3.24$ to\ $0.52$ days.
Meaning, interpretation same as in Sect.\ \@ref(PairedInvasivePlants).
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeanDiffTea).**
CI calculated as: $-49.36$ to $-27.88\mgs$.dL$^{-1}$.
Meaning, interpretation same as in Sect.\ \@ref(ChamomileTea-Paired).
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeanDiffGrowingSquash).**
**1.** *Analysis*: farm. *Observation*: individual fruits.
**2.** Pairs have same farm management, soil, etc.
**3.** Not shown.
**4.** Not shown.
**5.** Mean increase in fruit weight from normal (2015) to dry (i.e., normal minus dry). 
**6.** $H_0$: $\mu_d = 0$ and $H_1$: $\mu_d \ne 0$.
**7.** Not shown.
**8.** $t = 0.205$.
**9.** $P$ large; from software, $P = 0.839$.
**10.** $19.53\gs$ to $23.99\gs$ heavier in normal year (2015).
**11.** Probably stat.\ valid; $n$ just less than $25$.
**12.** No evidence ($t = 0.205$; two-tailed $P = 0.839$) of mean increase in weight of squash from dry to normal years (mean change: $2.23\gs$ ($95$%\ CI $-19.53$ to $23.99\gs$), heavier in normal year).
:::
```{r eval = FALSE}
2.23 + c(-2, 2)*10.88
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestPairedMeansCaptopril).**
**1.** Because it means blood pressure *reduction*; reduction is aim of drug.
**2.** Pairs have same sex, diet, age, etc.
**3.** The mean reduction in blood pressure.
**4.** Not shown.
**5.** $H_0$: $\mu_d = 0$ and $H_1$: $\mu_d > 0$ (because diffs are *reductions*).
**6.** Not shown.
**7.** $t = 8.12$.
**8.** $P = 0.001\div 2 = 0.0005$ (one-tailed).
**9.** $13.934$ to $23.93\mms$\ Hg from output.
**10.** $14.271$ to $23.56\mms$\ Hg.
**11.** Approx.\ CI uses *approx.* multipliers; exact uses exact multipliers.
**12.** Very strong evidence ($P = 0.0005$) drug reduces average systolic BP (mean reduction: $8.6\mms$\ Hg; $95$% CI: $13.934$ to $23.93\mms$\ Hg) in the pop.
**13.** $n = 14$; maybe not.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestPairedMeansTasteOfBroccoli).**
**1.** Not shown.
**2.** How much tastier broccoli is with dip.
**3.** $t = 1.699$; *approx.* one-tailed $P$ between $16$% and\ $2.5$%; not sure if $P$ larger than $0.05$, but likely ($t$-score quite a distance from $z = 1$).
       Evidence *probably* doesn't support $H_1$.
**4.** Approx. $95$%\ CI: $-0.92$ to\ $11.32$.
**5.** Stat.\ valid.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestPairedMeansSmokingAndExercise).**
**1.** $t = 1.78$; $P$ larger than $0.05$: evidence doesn't support $H_1$.
**2.** *Approx.* $95$%\ CI for *reduction*: $-0.08$ to\ $1.4$:.
**3.** Probably; $n$ just less than $25$.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestPairedMeansFerritin).**
$\bar{d} = -424.25$; $\text{s.e.}(\bar{d}) = 467.9404$; $n = 20$: $t = -0.907$.
$P = 0.376$: evidence doesn't support $H_1$.
$95$% CI: $-1403.7$ to\ $555.2$.
Test perhaps not stat.\ valid $n < 25$); but histogram of data suggests population *might* have normal distribution); $P$ so large probably makes little difference.
:::
```{r eval = FALSE}
data(Ferretin)
t.test(Ferretin$Reduction)
```




`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:StressSurgeryHT).**
**1.** $t = 2.48$; quite large; **one-tailed** $P$ certainly less than $0.025$.
       Strong evidence of population mean increase in beta-endorphin concentrations. 
**2.** $1.5$ to\ $13.9$.
**3.** $1.18$ to $14.22\fmols$/mol.
**4.** Approx.\ CI uses approx.\ multipliers.
**5.** $n = 19$, just less than $25$; results may not be stat.\ valid.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval = FALSE}
xbar <- 7.70
se <- 3.10
t <- xbar/se
P <- pnorm( -t )
c(t, P)
xbar + c(-2, 2)*se
```



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeanDiffCOVIDCI).**
**1.** *Diffs* are *during* minus *before*: *positive* diffs means *during* value higher.
**2.** $\text{s.e.}(\bar{d}) = 3.5150$.
**3.** $t = 0.762$; $P$ is large; no evidence of change; $-4.35$ to $9.71\mins$.
:::
```{r eval=FALSE}
xbar <- 2.68
se <- 51.30/sqrt(213)
t <- xbar/se
c(se, t, P = 2 * pnorm( -abs(t)) )
xbar + c(2, -2)*se
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeanDiffStudentEatingHabits).**
**1.** $0.63$ to\ $1.094\kgs$.
**2.** $0.63$ to\ $1.093\kgs$.
**3.** Very similar.
**4.** Yes.
**5.* $t = 7.431$; $P < 0.0001$; very strong evidence of change in mean weight gain.
      Probably not of practical importance.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:PairedCIExercisesAnorexia).**
$0.89$ to\ $4.64$ pounds.
Possibly not practically important.
:::
```{r eval=FALSE}
data(Anorexia)
with(Anorexia, 
     t.test(After - Before))
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:PairedCIExercisesSoilN).**
$t$ for mean reduction: $t = 2.41$; one-tailed $P = 0.0114$.
CI: $0.004$% to\ $0.052$%.
Strong evidence of mean difference.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval=FALSE}
data(SoilCN)
with(SoilCN, 
     t.test(IrrigatedN - NonirrigatedN))
```


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:PairedCIJumping).**
Exact CI: $1.204$ to\ $0.069\cms$ greater barefoot.
:::
```{r eval=FALSE}
data(Jumping)
with(Jumping, 
     t.test(Shoes - Barefoot))
```



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:PairedCIWCTennis}
Exact $95$%\ CI: $-0.01$ to\ $0.047\secs$ faster without racquet.
Perhaps not stat.\ valid.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval=FALSE}
data(WCTennis)
with(WCTennis, 
     t.test(PTwith - PTwithout))
```


::::::


```{r SBPHistDiffs, eval=includeGraphsTables, fig.cap="A histogram of the systolic blood pressure reductions (in mm Hg)", fig.align="center", fig.width=5, fig.height=3}
data(Captopril)

Captopril$Differences <- Captopril$Before - Captopril$After

bloodS <- subset(Captopril,
                 BP == "S")
bloodS <- bloodS[, c("Before",
                     "After",
                     "Differences")]

hist(bloodS$Differences,
     breaks = seq(0, 40, by = 10),
     main = "SBP reductions",
     xlim = c(-5, 35),
     las = 1,
     xlab = "SBP reduction (in mm Hg)",
     col = plot.colour)
box()
```




```{r FerritinGraph, eval=includeGraphsTables, fig.align="center", fig.cap="A histogram of the change in ferritin concentration", out.width='50%', fig.width=4, fig.height=2.5}
data(Ferritin)

hist(Ferritin$Reduction,
     breaks = seq(-4500, 6000, by = 1500),
     col = plot.colour,
     main = "Histogram of ferritin reduction",
     las = 1,
     ylim = c(0, 10),
     axes = FALSE,
     xlab = "Reduction (micrograms per litre)")
axis(side = 1,
     at = seq(-4500, 6000, by = 1500))
axis(side = 2,
     las = 1)

```





### Chap.\ \@ref(AnalysisTwoMeans): CIs and tests: comparing two means {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoSampleDiffsA).**
How much greater the mean lymphocytes cell diameter is compared to tumour cells.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoSampleDiffsB).**
How much greater mean braking distance is (worse) for cars with Type\ B brake pads compared to Type\ A pads.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoMeansCISamplingDistSignage).**
Normal; mean $\mu_B - \mu_A$; std\ dev.: $2.965\kms$.h^$-1$^.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoMeansCISamplingDistTea).**
Normal; mean at $\mu_T - \mu_W$; std\ dev.: $8.42\mgs.\xdLs^{-1}$.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoMeansWhales).**
**1.** Parameter: $\mu_F - \mu_M$.
       Estimate: $\bar{x}_F - \bar{x}_M = -0.06\ms$.
**2.** Not shown. <!-- See \@ref(fig:WhalesErrorbar). -->
**3.** $0.092487$
**4.** Not shown.
**5.** $-1.94$\ to $0.24\ms$. 
**6.** $H_0: \mu_F - \mu_M = 0$; $H_1: \mu_F - \mu_M \ne 0$.
**7.** $t = 0.65$; $P$ very large.
**8.** No evidence ($t = 0.65$; two-tailed $P > 0.10$) in sample that mean length of adult gray whales is diff in pop. for F (mean: $4.66\ms$; std dev.: $0..38\ms$) and M (mean: $4.60\ms$; std dev.: $0.30\ms$; $95$%\ CI for the diff: $-1.94\ms$ to $0.24\ms$).
**9.** Yes.
:::
```{r eval=FALSE}
se1 <- 0.38/sqrt(26)
se2 <- 0.30/sqrt(30)
se <- sqrt( se1^2 + se2^2)
xbar <- 4.66 - 4.60
xbar + c(-2, 2*se)
t <- xbar/se
c(t, 2 * pnorm(-abs(t)))
```

`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoMeansNHANES).**
**1.** Parameter: $\mu_N - \mu_YS$.
       Estimate: $\bar{x}_N - \bar{x}_Y = 0.085$ mmol.L^$-1$^.
**2.** Not shown.
**3.** Not shown.
**4.** $0.055$ to $0.115\mmols.\xLs^{-1}$.
**5.** $0.054$ to $0.115$ (bottom row).
**6.** $H_0$: $\mu_N - \mu_Y = 0$; $H_1$: $\mu_N - \mu_Y \ne 0$.
**7.** $0.015$.
**8.** $t = 5.478$ (second row); $P < 0.001$.
**9.** Very strong evidence to support $H_1$.
**10.** Yes.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeansIndSamplesExercisesEchinacea).**
**1.** $\mu_P - \mu_E$, reduction in mean duration for those using echinacea. 
**2.**  $0.2728678$ days; echinacea: $0.2446822$ days
**3.**  $0.3665054$.
**4.** Not shown.
**5.** $-0.203$ to\ $1.263$ days.
**6.** $5.85$ to $6.83$ days.
**7.** $H_0$: $\mu_P - \mu_E = 0$; $H_1$: $\mu_P - \mu_E > 0$ (one-tailed).
**8.** $0.3665054$.
**9.** $t = 1.45$; one-tailed $P$ between $0.025$ and $0.32$; using $z$-tables, $P$ approx. $0.074$.
**10.** Slight evidence of diff.
**11.** Not given.
**12.** Yes.
**13.** Probably not practically important (diff $0.53$ days).
:::
```{r eval=FALSE}
se1 <- 3.62/sqrt(176)
se2 <- 3.31/sqrt(183)
se <- sqrt(se1^2 + se2^2)
xbar <- 6.87 - 6.34
t <- xbar/se
c(t, pnorm( -abs(t)))
xbar + c(-2, 2)*se
6.34 + c(-2, 2)*se2
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:MeansIndSamplesExercisesCarpalTunnelSyndrome).**
No answer (yet).

**1.** Splinting *minus* exercise: diffs are how much greater pain is with splinting.
**2.** Exercise: $0.4427$; splinting: $0.3479$.
**3.** $0.563$
**4.** Not shown.
**5.** $-0.826$ to $1.426$: $0.826$ greater pain with exercise to $1.426$ greater pain with splinting.
**6.** $0.404$ to $1.796$.
**7.** $H_0$: $\mu_S - \mu_E = 0$; $H_1$: $\mu_S - \mu_E \ne 0$.
**8.** $t = 0.53$; very large $P$.
**9.** No evidence of difference.
**10.** Sample sizes small; probably not stat.\ valid: roughly correct only.
:::
`r if( !includeEvenAnswers) "-->"`

```{r eval=FALSE}
se1 <- 1.4/sqrt(10)
se2 <- 1.1/sqrt(10)
se <- sqrt(se1^2 + se2^2)
xbar <- 1.1 - 0.8
t <- xbar/se
c(t, pnorm( -abs(t)))
xbar + c(-2, 2)*se
1.1 + c(-2, 2)*se2
```

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoMeansDental).**
**1.** Amount of DMFT greater in non-industrialised countries, as upper table has negative mean.
**2.** $H_0$: $\mu_I - \mu_{NI} = 0$; $H_1$: $\mu_I - \mu_{NI} \ne 0$.
**3.** $11.9$ to $22.5$, greater for industrialised.
**4.** Very strong evidence in sample ($P < 0.001$) mean annual sugar consumption per person diff for industrialised (mean: $41.8\kgs$/person/y) and non-industrialised (mean: $24.6\kgs$/person/y) countries ($95$%\ CI for the diff $11.9$ to\ $22.5$).
**5.** Yes.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Deceleration).**
Perhaps: $\mu_{\text{After}} - \mu_{\text{Before}}$, *increase* in mean deceleration.
Approx. CI: $-0.025$ to $0.021\ms.\xsecs$.
$t = -0.175$; very large $P$; no evidence of increase.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval=FALSE}
xbar <- (-0.0020)
se <- 0.01143
se.me <- sqrt( (0.0494^2)/38 + (0.0521^2)/41 )
t <- xbar/se
c(t, pnorm( -abs(t)))
xbar + c(-2, 2)*se
```

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:FacePlant).**
**1.** $\mu_Y - \mu_O$: mean amount younger women can lean further forward than older.
**2.** Boxplot.
**3.** Not shown.
**4.** Approx.: $10.2$ to\ $18.8$^o^C.\spacex
       Exact CI (Row 2): $9.1$ to\ $19.9$^o^C.\spacex
       Different: sample sizes small.
**5.** One.
**6.** $H_0$: $\mu_Y - \mu_O = 0$; $H_1$: $\mu_Y - \mu_O > 0$.
**7.** $t = 6.69$ (*second row*); $P < 0.001/2$ as one-tailed; i.e., $P < 0.0005$.
**8.** Very strong evidence in sample ($t = 6.69$;  one-tailed $P < 0.0005$) that pop. mean one-step fall recovery angle for healthy women *greater* for young women (mean: $30.7$^o^C; std\ dev.: $2.58$^o^C; $n = 10$) compared to older women (mean: $16.20$^o^C; std\ dev.: $4.44$^o^C; $n = 5$; $95$%\ CI for the diff: $9.1$^o^C to $19.9$^o^C).
**9.** Probably not stat.\ valid.
:::
```{r eval=FALSE}
xbar <- 14.5
se <- 2.167
t <- xbar/se
c(t, pnorm( -abs(t)))
xbar + c(-2, 2)*se
```



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:BHADP).**
**1.**  Mean <span style="font-variant:small-caps;">bhadp</span> for those with disability, minus those without.
**2.** Not shown.
**3.** $0.78794$.
**4.** Not shown.
**5.** $5.18$ to $8.34$, larger for those with disability.
**6.** $H_0$: $\mu_D - \mu_N = 0$; $H_1$: $\mu_D - \mu_N \ne 0$ (one-tailed).
**7.** $t = \frac{6.76 - 0}{0.78794} = 8.58$.
**8.** $P$ *very* small.
**9.** Strong evidence exists in sample ($t = 8.58$; two-tailed $P < 0.001$) that people with disability (mean: $31.83$; $n = 132$; std dev.: $7.73$) and people without disability (mean: $25.07$; $n = 137$; std dev.: $4.80$) have diff pop. mean access to health promotion services ($95$%\ CI for diff: $5.18$ to\ $8.34$), as measured by <span style="font-variant:small-caps;">bhadp</span> scores.
**10.** Yes.
:::
`r if( !includeEvenAnswers) "-->"`

```{r eval=FALSE}
xbar <- 6.76
se <- sqrt(0.67280^2 + 0.41010^2)
t <- xbar/se
c(se, t, pnorm( -abs(t)))
xbar + c(-2, 2)*se
```


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestTwoMeansBodyTemperature).**
$H_0$: $\mu_M - \mu_{F} = 0$; $H_1$: $\mu_M - \mu_{F} \ne 0$.
From output, $t = -2.29$; (two-tailed) $P = 0.024$.
Moderate evidence ($P = 0.024$) mean internal body temperature diff for F (mean: $36.9$^o^C) and M (mean: $36.7$^o^C).
Diff between the means ($0.2$ of degree) of little *practical* importance.
:::
```{r eval=FALSE}
xbar <- -0.161
se <- 0.0703
t <- xbar/se
c(se, t, pnorm( -abs(t)))
xbar + c(-2, 2)*se
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestTwoMeansFitnessOfParamedics).**
**1.**: *grip*: $1.886$, $2.714$; *push-ups*: $2.357$, $3.317$.
**2.** $t = 1.51$; $P$ large; no evidence of diff.
**3.** $t = 2.70$; $P$ small; evidence of diff.
Sample sizes small; may not be stat.\ valid.
:::
`r if( !includeEvenAnswers) "-->"`

```{r eval=FALSE}
ses <- c(8, 9, 10, 11) / sqrt( c(18, 11, 18, 11))
mns <- c(51, 56, 36, 47)
sediffs1 <- sqrt( ses[1]^2 + ses[2]^2)
sediffs2 <- sqrt( ses[3]^2 + ses[4]^2)
c(sediffs1, sediffs2)

ts <- c( (51 - 56)/sediffs1,
         (36 - 47)/sediffs2 )
ts
pnorm( -abs(ts))

xbar + c(-2, 2)*se
```


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoMeansCIExercisesAnorexia).**
**1.** $2.76\kgs$.
**2.** CB: $0.227$ to $5.79\kgs$; Control: $-3.68$ to $2.78\kgs$. 
**3.** $-0.68$ to $7.59\kgs$, greater for CB.
**4.** $t = 1.67$; Two-tailed $P = 0.095$.
       Slight evidence of diff.
:::
```{r eval=FALSE}
data(Anorexia)
Anorexia$Gains <- with(Anorexia, 
                       After - Before)
df1 <- subset(Anorexia, 
              Anorexia$Treatment=="CB")
t.test(df1$Gain)
df2 <- subset(Anorexia, 
              Anorexia$Treatment=="Control")
t.test(df2$Gain)
se1 <- findStdError(df1$Gain)
se2 <- findStdError(df2$Gain)
sediff <- sqrt(se1^2 + se2^2)
c(se1, se2, sediff)
xbar <- (mean(df1$Gain) - mean(df2$Gains)) 
xbar + c(-2, 2)*sediff
t <- xbar/sediff
2 * pnorm( -abs(t))
```



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Coeliac).**
**1.** Researchers used exact $95$%CI.
**2.** True.
**3.** False.
**4.** $2.76$ means coeliacs have mean $2.76$ more DMFT.\spacex  
       $-2.32$ means *non*-coeliacs have a mean $2.32$ more DMFT.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoMeansReactionTimes).**
$t = 2.631$; one-tailed $P = 0.0055$; evidence of diff.
:::
```{r eval=FALSE}
data(ReactionTime)
with(ReactionTime,
     t.test(Reaction ~ factor(Group) ) )
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoMeansCIExercisesBMIanova).**
Table and graph not shown.
$F = 13.64$; $P < 0.001$; small.
Strong evidence that not all means are the same.
:::
`r if( !includeEvenAnswers) "-->"`
```{r eval=FALSE}
data(BMI)
with(BMI,
     anova( lm( exercise ~ factor(smoke) ) ) )
```
::::::



```{r NHANESSummaryTable, eval=includeGraphsTables}
NHANESSummaryTable <- array( NA,
                             dim = c(3, 4))
rownames(NHANESSummaryTable) <- c("Non-smokers",
                                  "Smokers",
                                  "Differences")
colnames(NHANESSummaryTable) <- c("Mean",
                                  "Standard deviation",
                                  "Standard error",
                                  "Sample size")
NHANESSummaryTable[1, ] <- c(1.3924,
                             0.42792,
                             0.01048,
                             1668)
NHANESSummaryTable[2, ] <- c(1.3077,
                             0.42353,
                             0.01137,
                             1388)
NHANESSummaryTable[3, ] <- c(0.08470,
                             NA ,
                             0.01546,
                             NA)


if( knitr::is_latex_output() ) {
  kable(pad(NHANESSummaryTable,
            surroundMaths = TRUE,
            targetLength = c(6, 7, 7, 4),
            decDigits = c(4, 5, 5, 0)),
        format = "latex",
        longtable = FALSE,
        escape = FALSE,
        col.names = colnames(NHANESSummaryTable),
        booktabs = TRUE,
        #linesep = c("\\addlinespace",  # Otherwise adds a space after five lines...
        align = c("r", "r", "r", "r"),
        caption = "A summary table for the \\textsc{nhanes} data; statistics in mm Hg") %>%
    column_spec(1, bold = TRUE) %>%
    row_spec(0, bold = TRUE) %>%
    kable_styling(font_size = 8)
}
if( knitr::is_html_output() ) {
  kable(pad(NHANESSummaryTable,
            surroundMaths = TRUE,
            targetLength = c(6, 7, 7, 4),
            decDigits = c(4, 5, 5, 0)),
        format = "html",
        longtable = FALSE,
        col.names = colnames(NHANESSummaryTable),
        booktabs = TRUE,
        align = "c",
        caption = "A summary table for the NHANES data; statistics in mm Hg") %>%
    kable_styling(full_width = FALSE) %>%
    column_spec(1, bold = TRUE)

}
```


```{r WhalesErrorbar, eval=includeGraphsTables, fig.cap="Error bar chart comparing the lengths of female and male gray whales", fig.align="center", fig.width=4.5, fig.height=3.25}

ciF <- c(4.66 - 2 * (0.379 / sqrt(26)),
         4.66 + 2 * (0.379 / sqrt(26)) )
ciM <- c(4.60 - 2 * (0.305 / sqrt(30)),
         4.60 + 2 * (0.305 / sqrt(30)) )


plot( range( c( ciF, ciM) ) ~ c(1, 2),
      type = "n",
      xlim = c(0.5, 2.5),
      ylim = c(4.45, 4.85),
      pch = 19,
      axes = FALSE,
      main = "Error bars chart:\n length of female gray whales at birth",
      xlab = "Sex",
      ylab = "Length (in metres)",
      sub = "(Error bars are 95% CIs)",
      las = 1)
axis(side = 1,
     at = 1:2,
     labels = c("Females", "Males"),
     las = 1)
axis(side = 2,
     las = 1)
box()

points( c(4.66, 4.60) ~ c(1:2),
        pch = 19,
        col = plot.colour,
        cex = 1.5)

arrows(1:2, c(ciF[1], ciM[1]),
       1:2, c(ciF[2], ciM[2]),
       length = 0.05,
       angle = 90,
       code = 3)
```



```{r FacePlantSummary, eval=includeGraphsTables}
data(ForwardFall)

FF.Summary <- array( NA, dim=c(3, 4))

FF.Summary[1:2, 1] <- aggregate(ForwardFall$LeanAngle ~ ForwardFall$Group,
                                FUN = "mean")[, 2]
FF.Summary[1:2, 2] <- aggregate(ForwardFall$LeanAngle ~ ForwardFall$Group,
                                FUN = "sd")[, 2]
FF.Summary[1:2, 3] <- aggregate(ForwardFall$LeanAngle ~ ForwardFall$Group,
                                FUN = function(x){sd(x) / sqrt(length(x))})[, 2]
FF.Summary[1:2, 4] <- aggregate(ForwardFall$LeanAngle ~ ForwardFall$Group,
                                FUN = "length")[, 2]

FF.Summary[3, 1] <- FF.Summary[1, 1] - FF.Summary[2, 1]
FF.Summary[3, 3] <- sqrt( FF.Summary[1, 2]^2/FF.Summary[1, 4] +
                            FF.Summary[2, 2]^2/FF.Summary[2, 4] )
row.names(FF.Summary) <- c("Younger women",
                           "Older women",
                           "Difference")

FF.Summary2 <- FF.Summary
FF.Summary2[, 1] <- round(FF.Summary[, 1], 1)
FF.Summary2[, 2] <- round(FF.Summary[, 2], 2)
FF.Summary2[, 3] <- round(FF.Summary[, 3], 2)
FF.Summary2[, 4] <- round(FF.Summary[, 4], 0)
FF.Summary2[3, 2] <- NA
FF.Summary2[3, 4] <- NA


if( knitr::is_latex_output() ) {
  kable(pad(FF.Summary2,
            surroundMaths = TRUE,
            targetLength = c(4, 4, 4, 2),
            decDigits = c(1, 2, 2, 0)),
        format = "latex",
        booktabs = TRUE,
        align = "r",
        escape = FALSE,
        col.names = c("Mean",
                      "Standard deviation",
                      "Standard error",
                      "Sample size"),
        caption = "Numerical summary for the face-plant data (means, standard deviations and standard errors are in degrees)"
  ) %>%
    row_spec(0, bold = TRUE) %>%
    kable_styling(font_size = 8)
}
if( knitr::is_html_output() ) {
  kable(pad(FF.Summary2,
            surroundMaths = TRUE,
            targetLength = c(4, 4, 4, 2),
            decDigits = c(1, 2, 2, 0)),
        format = "html",
        booktabs = TRUE,
        align = "c",
        col.names = c("Mean",
                      "Standard deviation",
                      "Standard error",
                      "Sample size"),
        caption = "Numerical summary for the face-plant data (means, standard deviations and standard errors are in degrees)"
  )
}
```




### Chap.\ \@ref(AnalysisOddsRatio): CIs and tests: comparing two odds or proportions {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OddsSame).**
Both odds: $6.04$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OddsSame2).**
Expected counts: $3.42$, $23.58$; $5.58$, $38.32$.
One less than five.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OddsRatioCISamplingDistA).**
Normal; mean $p_P - p_N$ and std\ dev. $0.0428$.
For OR: not normal distribution.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OddsRatioCISamplingDistB).**
Normal; mean $p_N - p_R$ and std\ dev. $0.1241$.
For OR: not normal distribution.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Chi2z).**
**1.** $z = 3.26$.
**2.** Very small.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Chi2zA).**
**1.** $z = 2.01$.
**2.** Just smaller than $0.05$.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:EVAdoption).**
**1.** No-PG proportion (Row\ 1), minus PG (Row\ 2).
**2.** $H_0$: $p_{\text{With}} - p_{\text{W'out}} = 0$; $H_1$: $p_{\text{With}} - p_{\text{W'out}} \ne 0$.
**3.** $z = 1.14$; $P = 0.253$; no evidence of diff.
**4.** $-0.071$ to\ $0.296$, larger intent to purchase for those without PG study.
**5.** Odds yes (Col\ 1), comparing no-PG (Row\ 1) to PG (Row\ 2).
**6.** One option: $H_0$: $\text{OR} = 1$; $H_1$: $\text{OR}\ne 1$.
**7.** $\chi^2 = 1.31$; $P = 0.253$; no evidence of diff.
**8.** OR between $0.68$, $4.28$.
**9.** Smallest expected: $10.6$; yes.
:::

`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:EthiopianFarmers).**
**1.** Proportion non-adopters in small (Row\ 1), minus proportion non-adopters in large (Row\ 2).
**2.** $H_0$: $p_{\text{S}} - p_{\text{L}} = 0$; $H_1$: $p_{\text{S}} - p_{\text{L}} \ne 0$.
**3.** $z = 3.33$; $P < 0.001$; evidence of diff.
**4.** $0.0881$ to\ $0.327$, larger proportion non-adopters in small farms.
**5.** Odds non-adopter, comparing small to large.
**6.** One option: $H_0$: $\text{OR} = 1$; $H_1$: $\text{OR}\ne 1$.
**7.** $\chi^2 = 11.10$; $P < 0.001$; no evidence of diff.
**8.** OR between $1.41$ and\ $3.84$.
**9.** Smallest expected count: $56.3$; yes.
:::
`r if( !includeEvenAnswers) "-->"`



```{r echo=FALSE, results='hide'}
p1 <- 15/(15 + 35)
p2 <- 37/(37 + 85)
p1 - p2

sep1 <- sqrt(p1 * (1 - p1)/ (15 + 35) )
sep2 <- sqrt(p2 * (1 - p2)/ (37 + 85) )
sediff <- sqrt(sep1^2 + sep2^2)

(p1 - p2) + c(-2, 2) * sediff


odds1 <- 15/35
odds2 <- 37/85
OR <- odds1 / odds2


pc <- (15 + 37)  / (15 + 37 + 35 + 85)
sepc1 <- sqrt(pc * (1 - pc)/ (15 + 35) )
sepc2 <- sqrt(pc * (1 - pc)/ (37 + 85) )
secdiff <- sqrt(sepc1^2 + sepc2^2)


smallestE <- (15 + 35) * (15 + 37) / (15 + 35 + 37 * 85)
```


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ORcrashes).**
**1.** Not shown.
**2.** $0.3000$; $0.3033$.
**3.** $-0.0033$.
**4.** $\text{s.e.}(\hat{p}_1) = 0.0648074$; $\text{s.e.}(\hat{p}_2) = 0.0416170$; s.e. for diff.: $0.077019$.
**5.** $-0.0033\pm 0.154$: $-0.157$ to\ $0.151$.
**6.** $-0.154$ to\ $0.148$.
**7.** Not given.
**8.** $0.429$; $0.436$.
**9.** $0.429/0.436 = 0.985$.
**10.** $0.480$ to\ $2.02$.
**11.** Not shown
**12.** $p = 0.4333$; s.e. for diff: $0.0832097$.
**13.** $z = (-0.0033 - 0)/0.0832097 = -0.040$; very small $P$; no evidence of diff.
**14.** $\chi^2 = 0.002$ (output); $P = 0.966$; no evidence of diff.
:::


```{r echo=FALSE, results='hide'}
p1 <- 216/(216 + 115)
p2 <- 99/(99 + 62)
p1 - p2

sep1 <- sqrt(p1 * (1 - p1)/ (216 + 115) )
sep2 <- sqrt(p2 * (1 - p2)/ (99 + 62) )
sediff <- sqrt(sep1^2 + sep2^2)

(p1 - p2) + c(-2, 2) * sediff


odds1 <- 216/115
odds2 <- 99/62
OR <- odds1 / odds2


pc <- (216 + 99)  / (216 + 99 + 115 + 62)
sepc1 <- sqrt(pc * (1 - pc)/ (216 + 115) )
sepc2 <- sqrt(pc * (1 - pc)/ (99 + 62) )
secdiff <- sqrt(sepc1^2 + sepc2^2)


smallestE <- (99 + 62) * (115 + 62) / (216 + 99 + 115 + 62)
```

`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestsOddsRatioScars).**
**1.** Not shown.
**2.** $0.653$; $0.615$.
**3.** $0.0377$.
**4.** $\text{s.e.}(\hat{p}_1) = 0.02617$; $\text{s.e.}(\hat{p}_2) = 0.03835$; s.e. for diff: $0.04643$.
**5.** $-0.0377\pm 0.0986$: $-0.055$ to\ $0.131$.
**6.** $-0.053$ to\ $0.129$.
**7.** Not given.
**8.** $1.878$; $1.597$.
**9.** $1.878/1.597 = 1.18$.
**10.** $0.80$ to\ $1.73$.
**11.** Not shown
**12.** $p = 0.640$; s.e. for diff: $0.04611$.
**13.** $z = (0.0377 - 0)/0.04611 = 0.818$; very small $P$; no evidence of diff.
**14.** $\chi^2 = 0.667$ (output); $P = 0.414$; no evidence of diff.
**15.** Smallest expected count: $57.9$; yes.
:::
`r if( !includeEvenAnswers) "-->"`



```{r echo=FALSE, results='hide'}
p1 <- 141/469
p2 <- 98/199
p1 - p2

sep1 <- sqrt(p1 * (1 - p1)/ 469)
sep2 <- sqrt(p2 * (1 - p2)/ 199)
sediff <- sqrt( sep1^2 + sep2^2)

(p1 - p2) + c(-2, 2) * sediff


pc <- (141 + 98) / (328 + 101)
sepc1 <- sqrt(pc * (1 - pc)/ 469)
sepc2 <- sqrt(pc * (1 - pc)/ 199)
secdiff <- sqrt( sepc1^2 + sepc2^2)

(p1 - p2) + c(-2, 2) * sediff

z <- (p1 - p2)/secdiff
```

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:PetBirdsTest).**
**1.** $-0.1918$.
**2.** $0.04643$.
**3.** $0.04202$.
**4.** Second uses common $p$: test assumes $p$ same in both groups.
**5.** $-0.1918 \pm (2\times 0.04643)$: $-0.285$ to\ $-0.099$.
**6.** $-0.273$ to\ $-0.111$.
**7.** Not given.
**8.** $z = (-0.1918 - 0)/0.04202 = -4.56$; small $P$; evidence of diff.
**9.** $0.443$.
**10.** $0.315$ to\ $0.623$.
**11.** $P < 0.0001$; small $P$; evidence of diff.
**12.** Yes.
**13.** Observational.
:::


```{r echo=FALSE, results='hide'}
p1 <- 53/(53 + 7)
p2 <- 40/(40 + 14)
p1 - p2

sep1 <- sqrt(p1 * (1 - p1)/ 60)
sep2 <- sqrt(p2 * (1 - p2)/ 54)
sediff <- sqrt( sep1^2 + sep2^2)

(p1 - p2) + c(-2, 2) * sediff

pc <- (53 + 40) / (53 + 7 + 40 + 14)
sepc1 <- sqrt(pc * (1 - pc)/ 60)
sepc2 <- sqrt(pc * (1 - pc)/ 54)
secdiff <- sqrt( sepc1^2 + sepc2^2)

(p1 - p2) + c(-2, 2) * sediff

z <- (p1 - p2)/secdiff
```

`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestsOddsRatioAugustRainfallInEmerald).**
**1.** $0.1426$.
**2.** $0.072622$.
**3.** $0.072715$.
**4.** Second uses common proportion, as test assume proportions same in both groups.
**5.** $0.1426 pm (2\times 0.072622)$: $-0.002$ to\ $-0.288$.
**6.** $0.000$ to\ $0.285$.
**7.** Not given.
**8.** $z = (0.1426 - 0)/0.072715 = 1.96$; marginal\ $P$; slight evidence of diff.
**9.** $2.65$.
**10.** $0.979$ to\ $7.17$.
**11.** $P = 0.050$; small $P$; slight evidence of diff.
**12.** Yes.
:::
`r if( !includeEvenAnswers) "-->"`



```{r echo=FALSE, results='hide'}
p1 <- 22/366
p2 <- 79/386
p1 - p2


odds1 <- 22 / (366 - 22)
odds2 <- 79 / (386 - 77)
odds1 / odds2

sep1 <- sqrt(p1 * (1 - p1)/ 366)
sep2 <- sqrt(p2 * (1 - p2)/ 386)
sediff <- sqrt( sep1^2 + sep2^2)

(p1 - p2) + c(-2, 2) * sediff

pc <- (22 + 79) / (366 + 386)
sepc1 <- sqrt(pc * (1 - pc)/ 366)
sepc2 <- sqrt(pc * (1 - pc)/ 386)
secdiff <- sqrt( sepc1^2 + sepc2^2)

(p1 - p2) + c(-2, 2) * sediff

z <- (p1 - p2)/secdiff
```


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestOddsRatioSunglasses).**
**1.** Not shown.
**2.** F: $0.060$, M: $0.205$; diff: $0.145$.
**3.** $0.0640$, $0.256$; $0.250$.
**4.** s.e. for diff: $0.0243$; $0.096$ to\ $0.193$.
**5.** $0.098$ to\ $0.192$.
**6.** Not given.
**7.** $2.45$ to\ $6.61$.
**8.** $P < 0.0001$.
**9.** Evidence of a diff.
**10.** Yes.
:::


```{r echo=FALSE, results='hide'}
p1 <- 7/73
p2 <- 9/42
p1 - p2


odds1 <- 7 / (73 - 7)
odds2 <- 9 / (42 - 9)
odds1 / odds2

sep1 <- sqrt(p1 * (1 - p1)/ 73)
sep2 <- sqrt(p2 * (1 - p2)/ 42)
sediff <- sqrt( sep1^2 + sep2^2)

(p1 - p2) + c(-2, 2) * sediff

pc <- (22 + 79) / (366 + 386)
sepc1 <- sqrt(pc * (1 - pc)/ 366)
sepc2 <- sqrt(pc * (1 - pc)/ 386)
secdiff <- sqrt( sepc1^2 + sepc2^2)

(p1 - p2) + c(-2, 2) * sediff

z <- (p1 - p2)/secdiff
```

`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:OddsRatiosCITurbinesHT).**
**1.** Not shown.
**2.** Diff: $-0.1184$; individual std errors: $0.03446$ and $0.063315$; for diff: $0.07208575$.
**3.** $0.389$.
**4.** $-0.263$ to\ $0.026$.
**5.** $-0.260$ to\ $0.023$.
**6.** Not given.
**7.** $0.133$ to\ $1.14$.
**8.** $P = 0/077$; slight evidence of diff.
**9.** Yes.
:::
`r if( !includeEvenAnswers) "-->"`




```{r echo=FALSE, results='hide'}
p1 <- 6 / (6 + 69)
p2 <- 15 / (15 + 60)
p1 - p2


odds1 <- 6 / 69
odds2 <- 15 / 60
odds1 / odds2

sep1 <- sqrt(p1 * (1 - p1)/ 75)
sep2 <- sqrt(p2 * (1 - p2)/ 75)
sediff <- sqrt( sep1^2 + sep2^2)

(p1 - p2) + c(-2, 2) * sediff

pc <- (6 + 15) / (6 + 15 + 69 + 60)
sepc1 <- sqrt(pc * (1 - pc)/ 75)
sepc2 <- sqrt(pc * (1 - pc)/ 75)
secdiff <- sqrt( sepc1^2 + sepc2^2)

(p1 - p2) + c(-2, 2) * sediff

sqrt(4.485)

z <- (p1 - p2)/secdiff
```

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TestOddsRatioBearTree).**
**1.** OR: $0.3478261$.
**2.** Diff: $-0.12$.
**3.** Proportions equal; not equal.
**4.** Odds equal; not equal.
**5.** $z = 2.12$; $P$ 'small'.
**6.** Some evidence of diff.
**7.** Yes.
:::

```{r echo=FALSE, results='hide'}
p1 <- 8 / (8 + 26)
p2 <- 8 / (8 + 82)
p1 - p2


odds1 <- 8 / 26
odds2 <- 8 / 82
odds1 / odds2

sep1 <- sqrt(p1 * (1 - p1)/ 34)
sep2 <- sqrt(p2 * (1 - p2)/ 90)
sediff <- sqrt( sep1^2 + sep2^2)

(p1 - p2) + c(-2, 2) * sediff

pc <- (8 + 8) / (8 + 8 + 26 + 82)
sepc1 <- sqrt(pc * (1 - pc)/ 34)
sepc2 <- sqrt(pc * (1 - pc)/ 90)
secdiff <- sqrt( sepc1^2 + sepc2^2)

(p1 - p2) + c(-2, 2) * sediff

sqrt(4.7067)

z <- (p1 - p2)/secdiff
```


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:B12Deficiency).**
**1.** OR: $3.15$.
**2.** Diff: $0.146$.
**3.** Proportions equal; not equal.
**4.** Odds equal; not equal.
**5.** $2.17$; small $P$.
**6.** Some evidence of diff.
**7.** $34\times 16/124 = 4.39$; *just* less than five; minor concern (should be noted).
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:DogsHT).**
**1.** $H_0$: No association; $H_1$: Association.
**2.** $23.0522$; $P = 0.00004$.
**3.** Very strong evidence of association.
**4.** Yes.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ComplianceHT).**
**1.** $H_0$: No association; $H_1$: Association.
**2.** $0.800$, $0.725$, $0.710$.
**3.** $4$, $2.642$, $2.444$.
**4.** $1.637$, $1.081$; reference level.
**5.** $1.0989$; $P = 0.577$.
**6.** No evidence of association.
**7.** Stat.\ valid.
:::
`r if( !includeEvenAnswers) "-->"`



```{r echo=FALSE, results='hide'}
odds <- c( 126/138,
           50/32,
           41/13)

odds[1] / odds[3]
odds[2] / odds[3]

pcent <- c(126 / (126 + 138),
           50 / (50 + 32),
           41 / (41 + 13) )


```

`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ShoppingBags).**
**1.** $0.913$, $1.56$, $3.15$.
**2.** $0.290$, $0.495$.
**3.** $47.7$%, $61.0$%, $75.9$%.
**4.** $H_0$: No association; $H_1$: Association.
**5.** $P$ small; evidence of association.
**6.** Yes.
:::
`r if( !includeEvenAnswers) "-->"`






`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CrabShells).**
**1.** $\chi^2 = 3.8764$; $P = 0.423$; no evidence of relationship between vertical, horizontal placement.
**2.** Smallest exp. count $0.67$; *not* stat.\ valid.
**3.** $\chi^2 = 0.237$; $P = 0.627$; no evidence of relationship between vertical, horizontal placement; stat.\ valid.
:::
`r if( !includeEvenAnswers) "-->"`




```{r BagsHTNumerical, eval=includeGraphsTables}
data(ShoppingBags)

ShoppingBags$BringBags <- ordered(ShoppingBags$BringBags, # Swap order
                                  levels = c("y", "n"))
BagsTab <- xtabs(Counts ~ AgeGroup + BringBags, 
                 data = ShoppingBags)

colnames(BagsTab) <- c("Brings own bags", 
                       "Does not bring own bags" )
rownames(BagsTab) <- c("30 and under",
                       "31 to 40",
                       "Over 40")


BagsSummary <- array( dim = c(3, 4) )

colnames(BagsSummary) <- c( "Odds",
                            "OR",
                           "Percentage",
                           "$n$")
rownames(BagsSummary) <- rownames(BagsTab)
  
BagsSummary[1, ] <- c( 0.913, 
                       0.289, 
                       47.7, 
                       264)
BagsSummary[2, ] <- c( 1.563, 
                       0.496, 
                       61.0, 
                       82)
BagsSummary[3, ] <- c(3.154, 
                      NA, 
                      75.9, 
                      54)
  
cap.tab <- "Odds and percentage that people bring their own shopping bags by age groups. The ORs are computed relative to those 'Over $40$'."


if( knitr::is_latex_output() ) {
  kable(pad(BagsSummary,
            surroundMaths = TRUE,
            targetLength = c(5, 5, 4, 3),
            decDigits = c(3, 3, 1, 0)),
        format = "latex",
        booktabs = TRUE,
        longtable = FALSE,
        escape = FALSE,
        align = "c",
        caption = cap.tab) %>%
    kable_styling(font_size = 7)
} else {
  kable(pad(BagsSummary,
            surroundMaths = TRUE,
            targetLength = c(5, 5, 4, 3),
            decDigits = c(3, 3, 1, 0)),
        format = "html",
        booktabs = TRUE,
        longtable = FALSE,
        align = "c",
        caption = cap.tab)
}
```












```{r PhoneTableExpected, eval=includeGraphsTables}
PhoneTable <- array( dim = c(2, 3))
PhoneTable[1, ] <- c(263, 259, 302)
PhoneTable[2, ] <- c(94, 98, 51)
colnames(PhoneTable) <- c("Answer call",
                          "Respond to text",
                          "Reply to email")
rownames(PhoneTable) <- c("Low exposure",
                          "High exposure")

if( knitr::is_latex_output() ) {
  kable(pad(round(chisq.test(PhoneTable)$expected, 2),
            surroundMaths = TRUE,
            targetLength = c(6, 6, 6),
            decDigits = 2),
        format = "latex",
        longtable = FALSE,
        booktabs = TRUE,
        escape = FALSE,
        align = "c",
        digits = 2,
        col.names = c("Answer call",
                      "Respond to text",
                      "Reply to email"),
        caption = "The expected counts for the phone-use data"
  ) %>%
    kable_styling(full_width = FALSE,
                  font_size = 8)
}
if( knitr::is_html_output() ) {
  out <- kable(pad(round(chisq.test(PhoneTable)$expected, 2),
            surroundMaths = TRUE,
            targetLength = c(6, 6, 6),
            decDigits = 2),
               format = "html",
            align = "c",
               longtable = FALSE,
               booktabs = TRUE,
               digits = 2,
               col.names = c("Answer call",
                             "Respond to text",
                             "Reply to email"),
               caption = "The expected counts for the phone-use data"
  ) %>%
    kable_styling(full_width = FALSE)
}
```



::::::



<!-- ```{r TurbinesORData} -->
<!-- tab.cap <- "The number of fissures for two sets of turbines, run for different numbers of hours" -->
<!-- TurbineCITable <- array(dim = c(3, 3) ) -->

<!-- TurbineCITable[1, ] <- c(7, 66, 73) -->
<!-- TurbineCITable[2, ] <- c(9, 33, 42) -->
<!-- TurbineCITable[3, ] <- c(16, 106, 122) -->

<!-- colnames(TurbineCITable) <- c("Fissures",  -->
<!--                               "No fissures", -->
<!--                               "Total") -->
<!-- rownames(TurbineCITable) <- c("About 1800 hours",  -->
<!--                               "About 3000 hours", -->
<!--                               "Total") -->

<!-- if( knitr::is_latex_output() ) { -->
<!--   kable(pad(TurbineCITable, -->
<!--             surroundMaths = TRUE, -->
<!--             targetLength = c(2, 3, 3), -->
<!--             decDigits = 0), -->
<!--         format = "latex", -->
<!--         longtable = FALSE, -->
<!--         col.names = colnames(TurbineCITable), -->
<!--         booktabs = TRUE, -->
<!--         escape = FALSE, -->
<!--         #linesep = c("\\addlinespace",  # Otherwise adds a space after five lines...  -->
<!--         align = "c", -->
<!--         caption = tab.cap) %>% -->
<!--     column_spec(1, bold = TRUE) %>% -->
<!--     kable_styling(font_size = 8) -->
<!-- } -->
<!-- if( knitr::is_html_output() ) { -->
<!--   out <- kable(pad(TurbineCITable, -->
<!--             surroundMaths = TRUE, -->
<!--             targetLength = c(2, 3, 3), -->
<!--             decDigits = 0), -->
<!--                format = "html", -->
<!--                longtable = FALSE, -->
<!--                col.names = colnames(TurbineCITable), -->
<!--                booktabs = TRUE, -->
<!--                align = "c", -->
<!--                caption = tab.cap) %>% -->
<!--     kable_styling(full_width = FALSE) %>% -->
<!--     column_spec(1, bold = TRUE)  -->
<!-- } -->
<!-- ``` -->




<!-- ```{r TurbinesORSummary} -->
<!-- tab.cap <- "The numerical summary for the fissures data" -->
<!-- TurbineCISumTable <- array(dim = c(3, 3) ) -->

<!-- TurbineCISumTable[1, ] <- c("0.1061",  -->
<!--                             "9.59%",  -->
<!--                             "73") -->
<!-- TurbineCISumTable[2, ] <- c("0.2727",  -->
<!--                             "21.43%",  -->
<!--                             "42") -->
<!-- TurbineCISumTable[3, ] <- c("0.389",  -->
<!--                             NA,  -->
<!--                             NA) -->

<!-- colnames(TurbineCISumTable) <- c("Odds with fissures", -->
<!--                                  "Percentage with fissures",  -->
<!--                                  "Sample size") -->
<!-- rownames(TurbineCISumTable) <- c("About 1800 hours",  -->
<!--                                  "About 3000 hours", -->
<!--                                  "Odds ratio") -->

<!-- if( knitr::is_latex_output() ) { -->
<!--   TurbineCISumTable[1, ] <- c("0.1061",  -->
<!--                               "$9.59$\\%",  -->
<!--                               "73") -->
<!--   TurbineCISumTable[2, ] <- c("0.2727",  -->
<!--                               "$21.43$\\%",  -->
<!--                               "42") -->
<!--   TurbineCISumTable[3, ] <- c("0.389",  -->
<!--                               NA,  -->
<!--                               NA) -->

<!--   kable(pad(TurbineCISumTable, -->
<!--             surroundMaths = TRUE, -->
<!--             targetLength = c(6, 4, 2), -->
<!--             decDigits = c(4, 2, 0)), -->
<!--         format = "latex", -->
<!--         longtable = FALSE, -->
<!--         col.names = colnames(TurbineCISumTable), -->
<!--         booktabs = TRUE, -->
<!--         escape = FALSE, -->
<!--         #linesep = c("\\addlinespace",  # Otherwise adds a space after five lines...  -->
<!--         align = "c", -->
<!--         caption = tab.cap) %>% -->
<!--     column_spec(1, bold = TRUE) %>% -->
<!--     kable_styling(font_size = 8) -->
<!-- } -->
<!-- if( knitr::is_html_output() ) { -->
<!--   out <- kable(pad(TurbineCISumTable, -->
<!--             surroundMaths = TRUE, -->
<!--             targetLength = c(6, 4, 2), -->
<!--             decDigits = c(4, 2, 0)), -->
<!--                format = "html", -->
<!--                longtable = FALSE, -->
<!--                col.names = colnames(TurbineCISumTable), -->
<!--                booktabs = TRUE, -->
<!--                align = "c", -->
<!--                caption = tab.cap) %>% -->
<!--     kable_styling(full_width = FALSE) %>% -->
<!--     column_spec(1, bold = TRUE)  -->
<!-- } -->
<!-- ``` -->



```{r ScarsNumericalSummary, eval=includeGraphsTables}
tab.cap <- "The odds and percentage of having smooth scars, for women and men"
ScarTable <- array(dim = c(3, 3) )
ScarTable[1, ] <- c("1.60",
                    "61.5%",
                    "161")
ScarTable[2, ] <- c("1.88",
                    "65.3%",
                    "331")
ScarTable[3, ] <- c("0.850",
                    NA,
                    NA)

colnames(ScarTable) <- c("Odds with smooth scars",
                         "Percentage with smooth scars",
                         "Sample size")
rownames(ScarTable) <- c("Women:",
                         "Men:",
                         "OR:")
if( knitr::is_latex_output() ) {
  ScarTable[1, ] <- c("1.60",
                      "$61.5$\\%",
                      "161")
  ScarTable[2, ] <- c("1.88",
                      "$65.3$\\%",
                      "331")
  ScarTable[3, ] <- c("0.850",
                      NA,
                      NA)

  kable(pad(ScarTable,
            surroundMaths = TRUE,
            targetLength = c(4, 4, 3),
            decDigits = c(2, 1, 0)),
        format = "latex",
        longtable = FALSE,
        escape = FALSE,
        col.names = colnames(ScarTable),
        booktabs = TRUE,
        #linesep = c("\\addlinespace",  # Otherwise adds a space after five lines...
        align = "c",
        caption = tab.cap) %>%
    column_spec(1, bold = TRUE) %>%
    kable_styling(font_size = 8)
}
if( knitr::is_html_output() ) {
  out <- kable(pad(ScarTable,
            surroundMaths = TRUE,
            targetLength = c(4, 4, 3),
            decDigits = c(2, 1, 0)),
               format = "html",
               longtable = FALSE,
               col.names = colnames(ScarTable),
               booktabs = TRUE,
               align = "c",
               caption = tab.cap) %>%
    kable_styling(full_width = FALSE) %>%
    column_spec(1, bold = TRUE)
}
```

```{r B12BarchartsCI, eval=includeGraphsTables, fig.cap="A side-by-side barchart comparing the number of women B12 deficient", fig.align="center", fig.width=4.5, fig.height=3, out.width="45%"}
### B12 example
counts <- matrix( c(8, 26, 8, 82),
                  byrow = TRUE,
                  nrow = 2)
rownames(counts) <- c("Veg.",
                      "Non-veg.")
colnames(counts) <- c("B12 def.",
                      "Not B12 def.")

mp <- barplot(t(counts),
        las = 1,
        ylab = "Number of women",
        col =  viridis::viridis(10)[c(3, 8)],
        legend.text = TRUE,
        ylim = c(0, 100),
        args.legend = list(x = "topleft",
                           bty = "n"),
        main = "Number of women who are\nand who are not B12 deficient",
        beside = TRUE)
box()
```



```{r B12DataSummaryCI, eval=includeGraphsTables}
B12summary <- array( dim = c(3, 3) )

colnames(B12summary) <- c( "Odds B12 deficient",
                          "Percentage B12 deficient",
                          "Sample size")
rownames(B12summary) <- c("Vegetarians:",
                          "Non-vegetarians:",
                          "OR:")


B12summary[1, ] <- c("0.3077",
                     "23.5",
                     "34")
B12summary[2, ] <- c("0.0976",
                     " 8.9",
                     "90")
B12summary[3, ] <- c("3.15",
                     NA,
                     NA)


if( knitr::is_latex_output() ) {
  kable(pad(B12summary,
            surroundMaths = TRUE,
            targetLength = c(5, 4, 2),
            decDigits = c(3, 1, 0)),
        format = "latex",
        booktabs = TRUE,
        longtable = FALSE,
        escape = FALSE,
        align = "c",
#      align=c("p{20mm}", "p{25mm}", "c"),
        caption = "The odds and percentage of subjects that are B12 deficient") %>%
    row_spec(0, bold = TRUE) %>%
    row_spec(3, italic = TRUE) %>%
    kable_styling(font_size = 8)
}
if( knitr::is_html_output() ) {
  kable(pad(B12summary,
            surroundMaths = TRUE,
            targetLength = c(5, 4, 2),
            decDigits = c(3, 1, 0)),
        format = "html",
        booktabs = TRUE,
        longtable = FALSE,
        align = "c",
        caption = "The odds and percentage of subjects that are B12 deficient")
}
```




```{r CrashDataSummary, eval=includeGraphsTables}
CrashSum <- array( dim = c(3, 3) )
colnames(CrashSum) <- c("Percentage involving pedestrians",
                        "Odds involving pedestrians",
                        "Sample size")
rownames(CrashSum) <- c("In 2011",
                        "In 2015",
                        "OR:")
CrashSum[1, ] <- c("30.0",
                   "0.429",
                   "50")
CrashSum[2, ] <- c("30.3",
                   "0.435",
                   "122")
CrashSum[3, ] <- c(NA,
                   0.985,
                   NA)

if( knitr::is_latex_output() ) {
  kable( pad(CrashSum,
             surroundMaths = TRUE,
             targetLength = c(4, 5, 3),
             decDigits = c(1, 3, 0)),
        format = "latex",
        booktabs = TRUE,
        longtable = FALSE,
        escape = FALSE,
        align = "c",
        caption = "Numerical summary of crashes in different years") %>%
    column_spec(1, width = "18mm") %>%
    column_spec(2, width = "32mm") %>%
    column_spec(3, width = "32mm") %>%
    column_spec(4, width = "17mm") %>%
     row_spec(0, bold = TRUE) %>%
     row_spec(3, italic = TRUE) %>%
     kable_styling(font_size = 8)
}
if( knitr::is_html_output() ) {
  kable(pad(CrashSum,
             surroundMaths = TRUE,
             targetLength = c(4, 5, 3),
             decDigits = c(1, 3, 0)),
               format = "html",
               booktabs = TRUE,
               longtable = FALSE,
               align = "c",
        caption = "Numerical summary of crashes in different years") %>%
    row_spec(0, bold = TRUE) %>%
     row_spec(3, bold = TRUE)
}
```



```{r BirdsNumericalSummary, eval=includeGraphsTables}
PBsummary <- array( dim = c(3, 3) )

colnames(PBsummary) <- c( "Odds not keeping pet bird",
                          "Percentage not keeping pet bird",
                          "Sample size")
rownames(PBsummary) <- c("With lung cancer:",
                         "Without lung cancer:",
                         "OR:")


PBsummary[1, ] <- c("1.439",
                    "59.0",
                    "238")
PBsummary[2, ] <- c("3.245",
                    "76.5",
                    "429")
PBsummary[3, ] <- c("0.443",
                    NA,
                    NA)


if( knitr::is_latex_output() ) {
  kable( pad(PBsummary,
             surroundMaths = TRUE,
             targetLength = c(6, 4, 3),
             decDigits = c(4, 1, 0)),
         format = "latex",
         align = "c",
         booktabs = TRUE,
         longtable = FALSE,
         escape = FALSE,
         col.names = c("pet birds",
                       "pet birds",
                       "size"),
         caption = "The odds and percentage of subjects keeping pet birds") %>%
    row_spec(0, bold = TRUE) %>%
    row_spec(3, italic = TRUE) %>%
    kable_styling(font_size = 8) %>%
    add_header_above( c(" " = 1,
                        "Odds not keeping"= 1,
                        "Percentage not keeping" = 1,
                        "Sample" = 1),
                      line = FALSE,
                      bold = TRUE)
}
if( knitr::is_html_output() ) {
  kable(pad(PBsummary,
             surroundMaths = TRUE,
             targetLength = c(6, 4, 3),
             decDigits = c(4, 1, 0)),
        format = "html",
        booktabs = TRUE,
        longtable = FALSE,
        align = "c",
        caption = "The odds and percentage of subjects keeping pet birds")
}
```



```{r BirdsGraphs, eval=includeGraphsTables, fig.cap="A plot of the pet-birds data", fig.align="center", fig.width=5.5, fig.height=3.5, out.width='55%'}
data(PetBirds)
PB2 <- xtabs( Counts ~ Pets + LC,
              data = PetBirds)

par(xpd = TRUE,
    mar = c(5, 4, 4, 8) + 0.1) # DEFAULT: c(5, 4, 4, 2) + 0.1

barplot(PB2,
        las = 1,
        ylab = "Count",
        xlab = "Lung cancer?",
        beside = TRUE,
        names.arg = c("With lung\ncancer",
                      "Without lung\ncancer"),
        ylim = c(0, 400),
        col =  viridis::viridis(10)[c(3, 8)]
)
box()

legend(6.2,
       400,
       fill =  viridis::viridis(10)[c(3, 8)],
       title = "Kept pets?",
       legend = c("Kept pet birds",
                  "Did not keep\npet birds"),
       bty = "n")
```


<!-- ::: {.answer data-latex=""} -->
<!-- **Ex.\ \@ref(exr:OddsRatiosCIEarInf).** -->
<!-- **1.** Individual std\ errors: $0.04191$; $0.04019$; diff: $0.05806$ -->
<!-- **2.** $-0.2927$ to\ $-0.0604$, or $0.0604$ to\ $0.2927$ greater for beach swimmers. -->
<!-- **3.** $0.0627$ to\ $0.2903$ greater for beach swimmers. -->
<!-- **4.** (No answer.) -->
<!-- **5.** Odds no infection: $61/79 = 0.772$ (non-beach) and $90/57 = 1.579$ (beach); OR: $0.490$ as in output. -->
<!-- **6.** $0.3054$ to $0.7831$ (Non-beach vs beach). -->
<!-- **7.** (No answer.) -->
<!-- **8.** $140\times136/287 = 66.3$; yes. -->
<!-- **9.** Not shown. -->
<!-- ::: -->





<!-- ::: {.answer data-latex=""} -->
<!-- **Ex.\ \@ref(exr:TestsOddsSandflies).** -->
<!-- **1.** Table not shown. -->
<!-- **2.** Graph not shown. -->
<!-- **3.** Moderate evidence that the *population* odds of finding a male sandfly in eastern Panama is different at $3$\ ft above ground  compared to $35$\ ft above ground. -->
<!-- **4.** Stat.\ valid. -->
<!-- ::: -->





### Chap.\ \@ref(EstimatingSampleSize): Finding sample sizes for CIs {.unlisted .unnumbered}


:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeNarrow1).**
Larger.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeNarrow2).**
Precise.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeMean1).**
**1.** At least $25$.
**2.** At least $100$ ($4$ times as many).
**3.** At least $400$ ($16$ times as many).
**4.** Halve width: $4$ times as many.
**5.** Quarter width: $16$ times as many.
**6.** More needed for greater precision.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeTwoMean2).**
**1.** At least $32$ in each.
**2.** At least $128$ ($4$ times as many).
**3.** At least $512$ ($16$ times as many.
**4.** Halve width: $4$ times as many units.
**5.** Quarter width: $16$ times as many units.
**6.** More needed for greater precision.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizePropEating).**
**1.** At least $10\,000$.
**2.** At least $2\,500$.
**3.** At least $1\,000$.
**4.** Expensive (time *and* money): $10\,000$ and $2\,500$ probably unrealistic.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeOneProportionAustSmokers).**
**1.** At least $n = 1/(0.05^2)  = 400$.
**2.** At least $n = 1/(0.025^2) = 1\,600$.
**3.** To *halve* width, $4$ times as many people needed.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeMeanLungCapacity).**
Use $s = 0.43$.
**1.** At least $1\,849$.
**2.** At least $296$.
**3.** At least $74$.
**4.** Expensive (time *and* money); $74$ more realistic.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeOneMeanBloodLossSampleSize).**
No answer (yet).
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeInvasivePlants).**
Use, say, $s = 13$.
**1.** At least $81$ pairs.
**2.** At least $76$ pairs.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeCaptopril).**
**1.** At least $81$.
**2.** At least $144$.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeWhales).**
Use, say, $s = 0.35$.
**1.** At least $44$ in each group.
**2.** At least $98$ in each group.
**3.** Info not relevant to goldfish.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizePneumonia).**
**1.** At least $13$ in each group.
**2.** At least $50$ in each group.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeHAttack).**
$2\,223$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SampleSizeSunglasses).**
$409$ M, $409$ F.
:::
`r if( !includeEvenAnswers) "-->"`


::::::





### Chap.\ \@ref(CorrelationRegression): Correlation and regression {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RegressionGuess).**
Answers *very* approximate.
**1.** $r$ moderate, positive; $\hat{y} = 4 + 1.5x$
**2.** $r$ reasonably strong, positive; $\hat{y} = 6 + 2.3x$.
**3.** $r$ not apt: variation in $y$ increases as $x$ increases.
**4.** $r$ reasonably strong, negative; $\hat{y} = 8 - 1.5x$.
:::
```{r eval=FALSE}
x <- runif(55,
           min = -2, 
           max = 4)

y1 <- 4 + 1.5 * x 
y2 <- 7 + 2.4 * x  
y4 <- 8.25 - 1.5 * x 
```

`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RegressionExerciseManifold).**
**1.** Too many decimal places. 
       $r$ not relevant: non-linear.
**2.** Inappropriate: non-linear.
**3.** Inappropriate; $\hat{y}$, not $y$; slope, intercept values *swapped*.
**4.** Very dodgy.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorrelationConsistency1).**
Any could be; can't tell.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorrelationConsistency2).**
**3.** and **4.** incorrect.
Either of others *may* be true.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RegressionValues).**
**1.** $b_0 = 3.5$; $b_1 = -0.14$.
**2.** $b_0 = 2.1$; $b_1 = -0.0047$.
**3.** $b_0 = -25.2$; $b_1 = -0.95$.
**4.** $b_0 = 0.15$; $b_1 = -0.22$.

:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RegressionValues2).**
**1.** $b_0 = -1.03$; $b_1 = 7.2$.
**2.** $b_0 = -0.46$; $b_1 = -1.88$.
**3.** $b_0 = 16$; $b_1 = 201$.
**4.** $b_0 = -0.032$; $b_1 = 3.04$.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:PlotAndPoints1).**
Not shown.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:PlotAndPoints2).**
Not shown.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorTestDrug).**
**1.** $H_0$: $\rho = 0$; $H_1$: $\rho \ne 0$.
**2.** No evidence of relationship.
**3.** Approx. linear; variation in STAI same for diff levels of experience.
$n \ge 25$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorTestPesticides).**
**1.** $H_0$: $\rho = 0$; $H_1$: $\rho \ne 0$.
**2.** Moderate evidence of relationship.
**3.**  $H_0$: $\rho = 0$; $H_1$: $\rho \ne 0$.
**4.** No evidence of  relationship.
**5.** Linearity; $n \ge 25$; constant variation in pesticide levels.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorrelationSoftdrink).**
No: variation increasing.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:TwoQuantExercisesMandibleTEST).**
Non-linear for larger gestational ages; variation not constant.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RegressionExerciseSunscreen).**
**1.** $b_0$: *no* time spent on application, mean $0.27\gs$ applied; nonsense.
       $b_1$: each extra minute adds average of $2.21\gs$ sunscreen.
**2.** Slope: g/min; intercept: g.
**3.** $\beta_0$ could be zero; makes sense.
**4.** $\hat{y} = 18\gs$.
**5.** $64$% reduction in unexplained variation using time.
**6.** $r = 0.8$; strong positive correlation.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RegressionPredictBirthWeight).**
**1.** Too many decimal places.
**2.** Probably; high, significant correlation.
**3.** *Intercept*: weight of infant with chest circumference zero; silly.
       *Slope*: average increase in birth weight (in g) for each $1\cms$ increase in chest circumference.
**4.** Intercept: cm; slope: cm/g.
**5.** $\hat{y} = 2\,538.7\gs$.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorTestDogs).**
**1.** Probably linear; increasing; approx. constant variance in $y$ as $x$ increases.
**2.** $H_0$: $\rho = 0$; $H_0$: $\rho > 0$.
**3.** $r = 0.837$; $P < 0.00005$.
       Very strong evidence of positive relationship.
**4.** Yes.
:::




`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorrelationExerciseSoil).**
**1.** Negative, moderate.
**2.** $R^2 = 67.07$%; unexplained variation in CBR reduces by about $67$% using PI.
**3.** From $-0.54$ to\ $-0.94$.
**4.** $P = 0.0001$; very strong evidence of relationship.
**5.** $n < 25$; may not be stat.\ valid.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Apnoea).**
**1.** $r = 0.264$.
**2.** $R^2 = 6.99$%; using neck circumference reduces unknown variation by about $7$%.
**3.** $\hat{y} = -24.47 + 1.36x$: $y$ is REI; $x$ is neck circum. (in cm).
**4.** Each $1\cms$ increase in neck circum. increases REI by average of $1.36$.
**5.** Approx. CI: $0.0575$ to\ $2.675$.
**6.** $t = 2.09$, $P = 0.041$: slight evidence of relationship.
**7.** Stat.\ valid.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:EDpatientsCI).**
**1.** $\hat{y} = 150.19 - 0.348x$ ($y$: mean number ED patients; $x$: number of days since welfare distribution).
**2.** Each extra day after welfare distribution associated with *decrease* in mean number of ED patients of about $0.35$.
       Easier: each $10$ extra days after welfare distribution associated with *decrease* in mean number of ED patients of about $10\times 0.35 = 3.5$.
**3.** $-0.441$ to\ $-0.255$ patients per day.
**4.** $t = -7.45$; two-tailed $P$ very small: $P < 0.00001$.
**5.** Negative; not sure of value.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorrelationRegressionExerciseBitumen).**
**1.** Very strong, negative linear relationship.
**2.** $r = -\sqrt{0.9929} = -0.9964$; must be negative.
**3.** $\hat{y} = 17.47 - 2.59x$: $x$ is percentage bitumen by wt; $y$ is percentage air voids by volume.
**4.** *Slope*: increase in bitumen wt by one percentage point *decreases* average percentage air voids by volume by $2.59$ percentage points.
       *Intercept*: extrapolation: $0$% bitumen content by wt, percentage air voids by volume $17.47$%.
**5.** $t = -74.9$: massive; extremely strong evidence ($P < 0.001$) of relationship.
**6.** $P < 0.001$, as for slope.
**7.** $\hat{y} = 4.5027$, or $4.5$%; good prediction, as relationship strong.
**8.** $\hat{y} = 1.909$, or $1.9$%; perhaps poor: extrapolation.
**9.** Yes.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorrelationRegressionExercisePossums).**
**1.** $r = \sqrt{0.2304} = 0.48$.
**2.** $\hat{y} = 110.6 + 0.021x$.
**3.** $t = 4.92$; $P < 0.001$; strong evidence of relationship.
**4.** $P < 0.001$, same as for slope.
**5.** Each extra m of elevation associated with mean increase in weight of $0.021\gs$.
**6.** $131.6\gs$; fine.
**7.** $114.8\gs$; extrapolation.
:::
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::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorrelationExercisesGorillas).**
**1.** $r = 0.271$; $R^2 = 7.3$%.
**2.** $t = 1.35$; $P = 0.190$; no evidence of relationship.
**3.** $\hat{y} = -5.647 + 0.123x$.
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:CorollaPrice).**
**1.** Approximately linear, negative, reasonably strong.
**2.** Condition; extras (air con, etc.); sedan/hatch; etc.
**3.** Expensive: above regression line.
**4.** Probably $3\,900.
**5.** $b_0\approx 20$; mean price of 2014 second-hand Corolla about $20\,000.
**6.** $b_1 = (2 - 20)/(14 - 0)\approx -1.3$; price *reduces* about $1\,300 each year older.
**7.** $\hat{y} = 20 - 1.3x$ *very* rough
**8.** Intercept: dollars; slope: dollars per year.
**9.** $\hat{y} = 16.41 - 0.96x$; $y$ price in thousands of dollars, $x$ age in years.
**10.** $r = -0.922$,\; $R^2 = 85$%.
**11.** $\hat{y} = 16.41 - (0.96\times 20) = -2.79$: extrapolating.
**12.** Hardly need a test:  $H_0$: $\beta_1 = 0$ vs $H_1$: $\beta_1 < 0$; $t = -13.71$, $P = 0.0001/2 = 0.00005$.
        Very strong evidence older cars fetch lower second-hand prices, on average.
**13.** $-0.958 \pm (2\times 0.070)$, or $-0.958\pm 0.140$, or $-1.098$ to $-0.818$.
**14.** Same, reflected left-to-right.
        (a)\ Size of $r$ same but positive: $r = 0.93$.
        (b)\ Same.
        (c)\ Same slope, but positive (both horizontal axis use values one year apart).
        (d)\ Intercept changes: the predicted value of the price if line is extended to year\ $0$; meaningless.
:::
`r if( !includeEvenAnswers) "-->"`


```{r echo=FALSE, results='hide'}
data(Elephants)
with( subset(Elephants, Sex=="A"),
      cor(Mass, Height))
with( subset(Elephants, Sex=="B"),
      cor(Mass, Height))
with( subset(Elephants, Sex=="A"),
      summary(lm(Mass ~ Height)) )
with( subset(Elephants, Sex=="B"),
      summary( lm(Mass ~ Height)) )
```

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ElephantsCor).**
**1.** Left probably F.
**2.** Sex\ B.
**3.** A: $r = 0.600$; B: $r = 0.815$.
**4.** B (M).
**5.** A: $\hat{y} = -2289 + 21.31x$; B: $\hat{y} = -3621 + 27.63x$.
**6.** Both: $P < 0 .0001$ (probably use one-tailed test). 
**7.** A: $2\,503\kgs$; B: $2\,596\kgs$.
**8.** Sample sizes, linearity OK; for Sex\ B, perhaps increasing variation.
:::





`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:JeansCor).**
**1.** Larger width associated with larger heights: small-to-moderate correlation.
**2.** $\rho$ possibly zero.
**3.** $\rho$ possibly zero.
**4.** Not shown.
**5.** Sex confounding variable; need separate reports for men's, women's.
:::
`r if( !includeEvenAnswers) "-->"`


```{r eval=FALSE}
data(Jeans)
mean(Jeans$MaxHeightFront, na.rm=TRUE)
mean(Jeans$MaxWidthFront, na.rm=TRUE)
tapply( Jeans$MaxHeightFront, list(Jeans$Sex), mean, na.rm=TRUE)
tapply( Jeans$MaxWidthFront, list(Jeans$Sex), mean, na.rm=TRUE)
```



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:RegressionDogsLife).**
$r = -0.712$ and $P < 0.001$.
Regression: $\hat{y} = 13.39 - 0.093x$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:GraphsTypingCor).**
Non-linear relationship.
:::
`r if( !includeEvenAnswers) "-->"`

::::::





### Chap.\ \@ref(SelectTest): Selecting an analysis {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionA).**
Only **3.**
**2.** Almost certainly; $n = 24$ very close to $n = 25$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:SamplingDistributionB).**
Only **2.**
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Method1).**
Summary of mean *diffs*; histogram of diffs.
Paired samples $t$-test; CI for mean diff.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Method2).**
Summary of two groups, plus diff between means; boxplot, error bar chart.
CI for diff between two means.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Method3).**
Comparing two proportions (or ORs); stacked, side-by-side bar chart.
CI for diff in proportions (or CI for OR).
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Method4}
Correlation or regression, if linear.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Method5).**
Correlation or regression, if linear.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:Method6).**
A summary of two groups (over $30$; $30$ and under), plus diff between means; boxplot, error-bar chart.
A $t$-test and CI for diff between two means.
:::
`r if( !includeEvenAnswers) "-->"`

::::::









### Chap.\ \@ref(WritingResearch): Writing and reporting research {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}

::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteWordChoice).**
**1.** *to*.
**2.** *its*.
**3.** One sample of $50$ individuals; use 'mean' or 'median', not 'average'; units of age is 'years'.
**4.** Should all be one sentence.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteWordChoice2).**
**1.** *their*.
**2.** 'Sample' is singular: sample *was* taken.
**3.** Effect.
**4.** Better than what? Better in what way (strength? durability?)
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteAmbiguous).**
**1.** Ambiguous; sound like cage is M; passive.
   'The cage contained one male rat.'
**2.** Seaweed removed from beaker, or from lake water?
   'The research assistant recorded the pH of the lake water (after removing weeds) in the beaker.'
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteAmbiguous2).**
**1.** Ambiguous; sounds like field was liquid.
   'Liquid fertiliser was applied to one of the fields.'
**2.** Diets do not lose weight; people lose weight; 'average' is vague.
   'The mean weight loss was greater on the new diet compared to the traditional diet.'
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteAmbiguous3).**
**1.**  'Substantial' if a *large* change is expected (quote statistics (e.g., $P$-value) if *statistically* significant intended).
**2.** 'The data *are*...'
:::



`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteAmbiguous4}
1. Correlation inappropriate for nominal data like sex.
2. 'Group' is singular.
:::
`r if( !includeEvenAnswers) "-->"`




::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteExercisesDecimals).**
Number decimal places ridiculous.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteExercisesLikelyToDie).**
'More likely' than what?
*Everybody* dies.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteExercisesStudent1).**
RQ: P, O, C and I unclear; fonts should be identified.
Perhaps better: for students, is mean reading speed for text in Georgia font same as for Calibri font?
**Abstract** statement poor (*fonts* are not fast or slow).
Perhaps: sample provided evidence mean reading speeds different ($P = ?$), when comparing text in Georgia font (mean: ?) and Calibri font (mean: ?; $95$%\ CI for diff: ? to\ ?).
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteExercisesStudent2).**
No units of measurement; jump-heights given to $0.001$ of cm.; table could also summarise information for each individual jump type; numerical summary shouldn't include $P$, $t$, or CI.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteExercisesStudent3).**
Variables *qualitative*: means inappropriate; use OR; values almost certainly refer to CI for OR.\spacex
Without more information, we can't be sure what the OR means.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:WriteExercisesStudent4).**
This study alone cannot *prove* anything; *difference* between hang times is of interest: appropriate CI is for difference between mean hang times.
:::
`r if( !includeEvenAnswers) "-->"`

::::::




### Chap.\ \@ref(Reading): Reading and critiquing research {.unlisted .unnumbered}

:::::: {.ChapAnswers data-latex=""}
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ReadExerciseiPhoneStepCounts).**
**1.** Convenience; self-selected. 
       Those in study *may* be different than those not in.
**2.** Inclusion criteria.
**3.** Ethical (drop-outs happen); accurate description of study.
**4.** Not ecologically valid.
**5.** Paired $t$-test.
**6.** Null: no mean diff between counts on phone, manually counted; alternative: a diff.
**7.** $P$ small; evidence mean diff in step-count between methods cannot be explained by chance: likely a diff.
**8.** Valid.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ReadExerciseHeadphones).**
**1.** Students at that uni (QUMS).
**2.** Observational.
**3.** Random method used (but small proportion did not participate): results should generalise to population (students at that uni).
**4.** Double-barrelled.
**5.** $\hat{p} = 0.8603$; $\text{s.e.}(\hat{p}) = 0;011781$; $0.837$ to\ $0.884$.
**6.** $t$-test comparing two means.
**7.** Null: mean HQL score same for F, M.
       Alt: different mean scores.
**8.** $\text{s.e.}(\text{diff}) = 0.2375$.
**9.** $t = -2.61$; expect small $P$; evidence mean diff for F, M.
**10.** $0.145$ to\ $1.085$, higher for M.
**11.** Yes.
**12.** Null: mean score same in three groups (no diff').
        Alternative: mean score *not* the same in three groups.
**13.** Only *ear*phone users.
**14.** Evidence mean not the same in three groups.
**15.** $19.6$ to\ $20.0$.
**16.** $0.20488$.
**17.** $t = 3.90$; small $P$, as in table.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ReadExerciseQuakes).**
**1.** Only some evidence of diff in mean age.
**2.** Comparing the two groups; age possible confounder.
**3.** Two-sample $t$-test.
**4.** $0.03376$.
**5.** $t = 2.07$; small $P$; evidence of diff.
**6.** Probably, given std errors rounded.
**7.** Conceptual.
**8.** Not shown.
**9.** $\chi^2$.
**10.** $z = 1.75$; $P$ between $5$% and\ $32$%: not helpful.
**11.** Observational: not cause-and-effect; no confounders noted; very restricted population.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ReadExerciseSelinium).**
**1.** Unexplained variation reduces by about $27$% using electrical conductivity.
**2.** $r = 0.52$.
**3.** Very strong evidence population slope not zero.
**4.** $\microgs.\xms.\dS^{-1}.\xLs^{-1}$.
**5.** Many correct forms; proportion of wells with Se concentration less than $2$\ $\mu$g.L^$-1$^ same for areas with, without Pliocene rocks within $5\kms$.
**6.** $5.61$.
**7.** Very small; very strong evidence proportion of wells with Se concentration less than  $2$\ $\mu$g.L^$-1$^ different for areas with, without Pliocene rocks within $5\kms$.
:::
`r if( !includeEvenAnswers) "-->"`



::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ReadExerciseLarva).**
**1.** $\chi^2$-test to compare proportions.
**2.** No evidence of diff in survival rates at temps.
**3.** Evidence surviving\ *Cx.* had larger mean size compared to surviving\ *Ae.*.
**4.** Two-sample $t$-test.
**5.** $0.010628$.
**6.** $t = 26.3$; very small $P$; very strong evidence of diff in mean lengths.
**7.** Yes.
**8.** Yes.
**9.** *Cx.*: evidence mean sizes at temps diff; *Ae.*: no evidence mean sizes at temps diff.
**10.** Two-sample $t$-tests.
**11.** Intercept: $-55.40$ to\ $16.28$; slope: $3.88$ to\ $59.40$.
**12.** $t = 2.28$; expect small $P$; evidence of linear association.
**13.** Need scatterplot to be sure, but $n \ge 25$.
**14.** When predator--size ratio increases by one, predation efficiency increases by $31.64$ percentage points.
**15.** Unknown variation reduces by $8.7$% using predator--size ratio.
**16.** $r = 0.294$.
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ReadExerciseMouth).**
**1.** No; patients may have tried to open their mouth wider than usual.
**2.** Evidence of relationship between height and MMO; moderate positive correlation (taller associated with larger MMO, in general).
**3.** $R^2 = 29$%; about $29$% of variation in MMO explained by height.
**4.** Slope: each extra $10\cms$ of height associated with $3.6\mms$ greater MMO, on average.
**5.** $\hat{y} = 54.29$.
**6.** Two-sample $t$-test.
**7.** Extremely small.
**8.** Need to see graph; $n \ge 25$.
**9.** Very strong evidence M have larger mean MMO than F.
**10.** Yes; relationship between height and MMO may be because M are taller on average.
**11.** Population quite restricted.
        Neither individuals nor assessors blinded.
        Non-random sample.
        May be others.
:::
`r if( !includeEvenAnswers) "-->"`


::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ReadExerciseTomatoes).**
**1.** Stratified?
**2.** Strong evidence the mean number of actinomycetes diff.
**3.** Possibly not; sample sizes small.
**4.** Very strong evidence mean number higher in CNV.
**5.** Larger actinomycetes numbers linearly associated with lower corky root severity.
**6.** $R^2 = 57.8$%; unknown variation decreases by $57.8$% using actinomycete abundance. 
:::


`r if( !includeEvenAnswers) "<!--"`
::: {.answer data-latex=""}
**Ex.\ \@ref(exr:ReadExerciseEyeMasks).**
**1.** Control, so any increase has a reference value.
**2.** Experimental.
**3.** Objective measures; aid managing placebo effect.
**4.** To ensure comparison groups included women of similar age (to help manage confounding).
**5.** $0.92208$.
**6.** $t = 0.542$.
**7.** Yes; no evidence of diff in mean age.
**8.** Yes.
**9.** To ensure comparison groups included women in similar rooms (to manage confounding).
**10.** Not shown.
**11.** $z = 0.592$; no evidence of diff in percentage of room with air-con.
**12.** Yes.
**13.** Yes.
**14.** Paired $t = 8.93$; very small $P$; very strong evidence of mean increase.
**15.** Paired $t = 5.33$; small $P$; very strong evidence of mean increase.
**16.** Hawthorne effect.
**17.** $\text{s.e.}( \bar{x}_T - \bar{x}_T) = 4.378$.
**18.** $-2.16$ to $15.36\mins$, greater for treatment group.
        $t = 1.51$; one-tailed $P$ not very small; no diff between mean increase in sleep times.
**19.** Yes.
:::
`r if( !includeEvenAnswers) "-->"`


::::::

<!-- Restore size of captions -->
`r if( knitr::is_latex_output() ) "\\captionsetup{font=normalsize}"`