# E Answers to end-of-chapter exercises

### Chap. 1: Introduction

### Chap. 2: RQs

**Ex. 2.1.**
**1.** *Percentage* of vehicles that crash.
**2.** *Average* jump height.
**3.** *Average* number of tomatoes per plant.

**Ex. 2.3.**
**1.** Type of diet.
**2.** Whether coffee is caffeinated or decaffeinated.
**3.** Number iron tablets per day.

**Ex. 2.5.**
**1.** *Between*-individuals.
Outcome: percentage wearing hats.
**2.** *Between*-intervals.
Outcome: average yield (in kg/plant, tomatoes/plant, etc).

**Ex. 2.7.**
**1.** Correlational.
**2.** No sense assigning one variable as explanatory, another response.

**Ex. 2.9.**
**1.** P: Danish University 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? full-time?). *Operational*: how 'resting diastolic blood pressure' measured.
**6.** Resting diastolic blood pressure; whether they regularly drive, ride to uni.
**7.** Danish university students; Danish university students.

**Ex. 2.11.**
**1.** Probably relational.
**2.** Two-tailed.
**3.** Probably not.
**4.** How *individual* people using phones ('Talking'; 'texting').
**5.** Walking speed.
**6.** *Average* walking speed.

**Ex. 2.13.**
**1.** Animal.
**2.** *Pen*: food is allocated to pen.
Animals in the same pen are not independent: they compete for the same space, food, resources, and have similar environments.
**3.** Between diets.

**Ex. 2.15.**
The \(10\) adults is sample.
Unclear how many fonts compared (or which fonts).
Perhaps: 'Among Australian adults, is the average time taken to read a passage of text different when Arial font is used compared to Times Roman font?'

**Ex. 2.17.**
**1.** Analysis: person; observation: individual nose hairs.
Each unit of analysis has \(50\) units of observation.
**2.** \(n = 2\).

**Ex. 2.19.**
**1.** P: American adults; individuals: American adults.
**2.** O: average number of recorded steps.
**3.** Response: number of steps recorded for individuals.
Explanatory: location of accelerometer.
**4.** *Within* individuals.

**Ex. 2.21.**
**1.** Relational; decision-making.
**2.** Correlational; estimation.

Unclear if intervention; seems unlikely.

**Ex. 2.23.**
Unit of observation: tyre.
Unit of analysis: car.
Brand allocated to car; each car gets only the same brand of tyre.
Tyres on one car exposed to the same day-to-day use, drivers, distances, conditions, etc.

Each unit of analysis (car) produces four units of observations.
*Sample size*: \(10\) cars (\(40\) observations).

**Ex. 2.25.**
The board.
Five units of analysis.
Ten.
Ten.
Within-board variation much smaller (apart from first board).

### Chap. 3: Overview of research design

**Ex. 3.1.**
**1.** Arsenic concentration.
**2.** Distance of lake from mine.
**3.** No: recorded; cannot be lurking.
**4.** Yes: may be related to the response, explanatory variables.
**5.** Confounding variable.
**6.** Observational: researchers do not determine the distance of lakes from mine.

**Ex. 3.3.**
*Response*: perhaps 'risk of developing a cancer of the digestive system'.
*Explanatory*: 'whether or not the participants drank green tea at least three times a week'.
*Lurking*: 'health consciousness of the participants' (appears unrecorded).

**Ex. 3.5.**
True; false; false; false; false; true.
False.
True.

### Chap. 4: Types of study designs

**Ex. 4.1.**
**1.** Between-individuals.
**2.** Relational.
**3.** Most likely.
**4.** Estimation.
**5.** Intervention, so experiment.
Likely true experiment.

**Ex. 4.3.**
*True experiment*.

**Ex. 4.5.**
*Quasi-experiment*.

**Ex. 4.7.**
**1.** Many answers possible.
**2.** Researchers *intervene*: researchers *give* or *not give* subjects a pet.
**3.** Researchers *do not intervene*: find the subjects who *do* or *do not* already own a pet.

### Chap. 6: Sampling

**Ex. 6.1.**
c. Externally-valid study more likely.

**Ex. 6.3.**
**1.** Every \(7\)th day is same day of week.
**2.** Maybe select days at random over three-months.

**Ex. 6.5.**
**1.** Multi-stage.
**2.** Stratified (selecting floor), then convenience.
**3.** Convenience.
**4.** Part stratified (selecting floors), then convenience.
First might be best.

**Ex. 6.7.**
Random sampling to select schools.
Then, self-selecting.

**Ex. 6.9.**
Stratified: zones are strata.

**Ex. 6.11.**
In Stage \(2\), selection of farms *not* random.

**Ex. 6.13.**
**1.** Households in Santiago.
**2.** ...if the sample is representative of all households in Santiago.
**3.** Voluntary response.
**4.**Multi-stage.

### Chap. 7: Internal validity

**Ex. 7.1.**
All are false.

**Ex. 7.3.**
Yes; yes; yes; yes; yes; no (*external* validity).

**Ex. 7.5.**
Also possible in observational studies.

**Ex. 7.7.**
Probably in case hive size is a confounder.

**Ex. 7.9.**
Statements 1, 3, 4, 8 and 9 true.
'Sex', 'Initial weight' possible confounders.

**Ex. 7.11.**
**1.** Observational.
**2.** *Response*: amount of sunscreen used; *explanatory*: time applying sunscreen.
**3.** Potential confounding variables.
**4.** If the mean of both the response and explanatory variables was different for females and males, sex of the participant a *confounder*; would need to be factored into the data analysis.
**5.** Participants blinded to what is happening in study.

**Ex. 7.13.**

**1.** Randomly allocate type of water to subjects (or the *order* subjects taste *each* drink.)
**2.** Subjects do not know which type of water they are drinking.
**3.** Person providing water and receiving ratings does not know which type of water subjects drinking.
**4.** Hard to find a control.
**5.** Any random sampling is good, if possible.

Observer effect: researcher directly contacting the subjects; may unintentionally influence responses.

**Ex. 7.15.**
Carry-over effect; observer effect.

### Chap. 8: Research design limitations

**Ex. 8.1.**
External.

**Ex. 8.3.**
Population: 'on-campus university students where (I) work'.
External validity: whether the results apply to other members of *target population*.

**Ex. 8.5.**
Sample not random; the researchers (rightly) state that results may not *generalise* to all hospitals.
Because data only collected at night, perhaps not *ecologically valid*.

**Ex. 8.7.**
Observational study: people with severe cough may take more cough drops.

**Ex. 8.9.**
Study lacks *ecological validity*.

### Chap. 9: Collecting data

**Ex. 9.1.**
No place for \(18\)-year-olds.

**Ex. 9.3.**
Best: second.
First: *leading* (*concerned* cat owners...)
Third: *leading* (Do you *agree*...)

**Ex. 9.5**.
First fine; 'seldom' (for instance) may mean different things to different people; possible recall bias.
Second: overlapping options (both \(1\) h and \(2\) h in two categories).

### Chap. 10: Classifying data

**Ex. 10.1.**
Quant. continuous.
Qual. nominal.
Quant. continuous.
Qual. nominal.

**Ex. 10.3.**
False; true; false

**Ex. 10.5.**
Nominal; qualitative.

**Ex. 10.7.**
**1.** Blood pressure: quant. continuous.
**2.** Program: qual. nominal.
**3.** Grade: qual. ordinal.
**4.** Number of doctor visits: quant. discrete.

**Ex. 10.9.**
*Gender*: qual. nominal.
*Age*: quant. continuous.
*Height*: quant. continuous.
*Weight*: quant. continuous.
*GMFCS*: qual. ordinal.

**Ex. 10.11.**
*Kangaroo response*: qual. ordinal (perhaps nominal?).
*Drone height*: quant.; with four values used; probably treated as qual. ordinal.
*Mob size*: quant. discrete.
*Sex*: qual. nominal.

### Chap. 11: Summarising quantitative data

**Ex. 11.1.**
Average: perhaps \(70\)--\(80\)?
Variation: most between \(30\) and \(80\).
Shape: skewed *left*.
Outliers: none; 'bump' at lower ages.

**Ex. 11.3.**
Average around \(1.5\) mmol/L.
Most between \(4\) and \(3\) mmol/L.
Slightly right skewed.
Some large outliers.

Probably the median as slightly skewed right, but with some outliers.
*Both* the mean and median *can* be quoted.

**Ex. 11.5.**
**1.** \(3.7\).
**2.** \(3.5\).
**3.** \(1.888562\).

**Ex. 11.7.**
Stemplot not shown.
**1.** Mean: \(-2.42\).
**2.** Median: \(0.8\).
**3.** Range: \(29.6\) (from \(-19.8\) to \(9.8\)).
**4.** Std. dev.: \(9.831172\); about \(9.83\).
**5.** IQR: \(4.95 - (-11.4) = 16.35\) (*not* including the median in each half).
(No units of measurement.)

**Ex. 11.9.**
**1.** In order (in cm): \(127.4\); \(129.0\); \(14.4\); \(24\) using my software.
Manually (*without* median in each half): \(Q_1 = 113\) and \(Q_3 = 138\) so IQR is \(25\).
**2.** We don't know.
**3.** No answer.
**4.** No answer.
**5.** No answer.
**6.** Hard to describe with standard language.

**Ex. 11.11.**
No answer (yet).

**Ex. 11.13.**
D; C; A; D.

### Chap. 12: Summarising qualitative data

**Ex. 12.1.**
Most common social group: many females plus offspring.
No commonly-observed social group include males.
Graph not shown.

**Ex. 12.3.**
None are *bad*.
I'd prefer bar chart, but any OK.

**Ex. 12.5.**
**1.** Nominal: gender; ordinal: place of residence; responses.
**2.** Gender: modes are F and M. Place: City \(> 100\ 000\) residents. Response: Agree.
**3.** Gender: NA. Place: City \(20\ 000\) to \(100\ 000\) residents. Response: Neutral.
**4.** \(5.12\): respondents about five times more likely to come from city than rural.
**5.** \(0.613\): respondents about \(0.61\) times as likely to agree or strongly disagree than choose other option.
**6.** \(1\): respondents just as likely to be male as female.

**Ex. 12.7.**
**1.** Walking; Bus
**2.** Bus.
**3.** No.
**4.** \(3.44\); that is, students \(3.44\) times as likely to use motorised transport than active transport.
**5.** \(0.141\); that is, for every \(100\) students that *do not walk* to campus, about \(100\times 0.141 = 14.1\) *do walk* to campus.
**6.** Figure not shown.
The left panel shows the specific methods, and the right panel shows the methods of transport grouped more coarsely.

**Ex. 12.9.**
*Age*, *FEV* and *Height*: histogram.
*Gender*: bar or dot; mode: male (\(51.4\)%; odds: \(1.06\)).
*Smoking*: bar or pie; mode: non-smoking (\(9.9\)%; odds: \(0.11\)).

### Chap. 13: Qualitative data: Comparing between individuals

**Ex. 13.1.**
**1.** *Vomited*: \(0.50\) beer then wine; \(0.50\) wine only.
*Didn't vomit*: \(0.738\) beer then wine, \(0.262\) wine only.
Prop. that drank various things, among those who did and didn't vomit.
**2.** *Beer then 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.** \(0.096774\).
**6.** \(2.82\).
**7.** \(0.354\).
**8.** \(-0.176\).
**9.** Higher percentage vomited after beer-then-wine, compared to beer only.

**Ex. 13.3.**
**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.

**Ex. 13.5.**
Plot not shown.

**Ex. 13.7.**
**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.** Odds ratio: \(1.41\).
**6.** Odds females skipping are \(1.41\) *times* the odds males skipping.
**7.** Not shown.

**Ex. 13.9.**
\(39.1%\).
\(28.7\)%.
\(0.642\).
\(0.402\).
\(1.6\).
Table not shown.

### Chap. 14: Quantitative data: Comparing within individuals

### Chap. 15: Quantitative data: Comparing between individuals

**Ex. 15.1.**
The DB method, in general, produces smaller cost over-runs.

**Ex. 15.3.**
**A.** II (median; IQR).
**B.** I (mean; standard deviation).
**C.** III (median; IQR).

**Ex. 15.5.**
Plot not shown.

**Ex. 15.7.**
No answer (yet).

**Ex. 15.9.**
**1.** Table not shown.
**2.** Plot not shown.

**Ex. 15.11.**
**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`

).
**2.** Table not shown.
**3.** Little difference between males, females in *sample*.

**Ex. 16.3.**
You cannot be very precise.
From software: \(r = 0.71\).
The best you can do is 'a reasonably high, positive \(r\) value'.

**Ex. 16.5.**
**1.** *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 about \(60\)).
**2.** Each point is a tree.

**Ex. 16.7.**
Relationship prob. linear... some top-right observations look different.
Variation increase a bit as Age increases.
Observations in top right seem to not follow the linear relationship.

**Ex. 16.9.**
Non-linear; higher wind speed related to higher DC output (in general); small to moderate variation.
DC output increases as wind speed increases, but not linearly.

**Ex. 16.11.**
No answer (yet).

### Chap. 17: More about summarising data

**Ex. 17.1.**
Scatterplot; histogram of the diffs; side-by-side bar.

**Ex. 17.3.**
Not shown.

**Ex. 17.5.**
Plotting symbols unexplained.
Axis labels unhelpful.
Vertical axis could stop at \(20\).

**Ex. 17.7.**
**1.** Response: *change* in MADRS (quant. continuous).
**2.** Explanatory: treatment group (qual. nominal, three levels).
**3.** Response: histogram. Explanatory: bar chart. Relationship: boxplot.

**Ex. 17.9.**
Plots not shown.
*Speed*: average: around \(60\) wpm; variation: about \(30\) to about \(120\) wpm.
Slightly right skewed; no obvious outliers.
*Accuracy*: average: around \(85\)%; variation: about \(65\)% to about \(95\)%.
Left skewed; no obvious outliers.
*Age*: average: \(25\); variation: about \(15\) to \(35\).
*Very* right skewed, perhaps large outliers we cannot see.
*Sex*: about twice as many females as males.
*Speed* and *Sex*: not big difference between M and F.
*Accuracy* and *Age*: hard to see relationship; no older people are 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?

### Chap. 18: Probability

**Ex. 18.1.**
**1.** Subjective.
**2.** Rel. frequency.

**Ex. 18.3.**
False.
True.
\(1/2\).
\(1/2\).

**Ex. 18.5.**
\(4/6\).
\(5\).
Yes: what happens on die won't change coin outcome.
\(1/2\).
\(1/6\).
\(1/3\).

**Ex. 18.7.**
No answer (yet).

**Ex. 18.9.**
**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.
A positive test result may or may not mean the person has the disease.
**4.** \(99/117\), or \(84\)% of having disease.

**Ex. 18.11.**
Events not equally likely.

### Chap. 19: Making decisions

### Chap. 20: Sampling variation

### Chap. 21: Distributions and models

**Ex. 21.1.**
*All* false.

**Ex. 21.3.**
**1:** C; **2:** A; **3:** B; **4:** D.

**Ex. 21.5.**
**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.** \(z\)-score about \(1.04\); tree diameter about \(11.6\) inches.

**Ex. 21.7.**
**1.** \(z = -0.61\); about \(72.9\)%.
**2.** \(z = -1.83\); about \(3.4\)%.
**3.** \(z = -4.878\) and \(z = -1.83\); about \(3.4\)%.
**4.** \(z = 1.64\) (or \(1.65\)).
\(5\)% *longer* than about \(42.7\) weeks.
**5.** \(z\)-score: \(-1.28\).
\(10\)% *shorter* than about \(37.9\) weeks.

**Ex. 21.9.**
Lower than about \(80.8\): rejection.

**Ex. 21.11.**
\(-1.77\); \(10\); mass lower than average of \(14\) tonnes.

### Chap. 23: CIs for one proportion

**Ex. 23.1.**
\(\hat{p} = 0.81944\) and \(n = 864\).
\(\text{s.e.}(\hat{p}) = 0.01309\); approx. \(95\)% CI: \(0.819 \pm (2\times 0.0131)\).
Stat. valid.

**Ex. 23.3.**
\(\hat{p} = 0.05194805\); \(\text{s.e.}(\hat{p}) = 0.0017833\); approx. \(95\)% CI: \(0.0519\pm 0.0358\).
Stat. valid.

**Ex. 23.5.**
\(\hat{p} = 0.317059\); \(n = 6882\).
\(\text{s.e.}(\hat{p}) = 0.005609244\).
CI: \(0.317\pm 0.011\).
Stat. valid.

**Ex. 23.7.**
\(\hat{p} = 365/1516 = 0.241\).
\(\text{s.e.}(\hat{p}) = 0.01098449\).
Approx. \(95\)% CI: \(0.219\) to \(0.263\).

### Chap. 24: CIs for one mean

**Ex. 24.1.**
**1.** *Parameter*: population mean weight of an American black bear, \(\mu\).
**2.** \(\text{s.e.}(\bar{x}) = 3.756947\).
**3.** \(77.4\) to \(92.4\) kg.
**4.** Approx. \(95\)% confident the *population mean* weight of male American black bears between \(77.4\) and \(92.4\) kg.
**5.** Stat. valid: \(n \ge 25\).

**Ex. 24.3.**
\(\text{s.e.} = 0.06410062\).
*Approx.* \(95\)% CI: \(2.72\) L to \(2.98\) L.

**Ex. 24.5.**
Approx. \(95\)% CI : \(29.9\) s to \(36.1\) s.

**Ex. 24.7.**
*None* acceptable.
**1.** CIs not about observations (nor parameters), but *statistics*.
**2.** CIs not about observations, but *statistics*.
**3.** *Samples* don't vary between two values; *statistics* vary.
(And CIs are about populations, not samples.)
**4.** *Populations* can't vary between two values.
**5.** Parameter do not vary.
**6.** We know \(\bar{x} = 1.3649\) mmol/L.
**7.** We know \(\bar{x} = 1.3649\) mmol/L.

**Ex. 24.9.**
\(\text{s.e.}(\bar{x}) = 5.36768\); approx. \(95\)% CI: \(50.56\) s to \(72.04\) s.
Stat. valid.

### Chap. 25: More about CIs

### Chap. 26: CIs for paired data

**Ex. 26.1.**
**1.** Paired.
**2.** Paired.

**Ex. 26.3.**
**1.** *Unit of analysis*: farm. *Units of observation*: individual fruits.
2. Table not shown.
3. Plot not shown.
4. The mean increase in average fruit weight from 2014 (dry year) to 2015 (normal year) is \(2.230\) g (\(\text{s.e.} = 10.879\); \(n = 23\)), with an approx. \(95\)% CI between \(24.79\) g lighter in 2014 to \(20.33\) g higher in 2015.

**Ex. 26.5.**
Mean of *diffs*: \(5.2\); std error: \(3.6\).
Approx. \(95\)% CI: \(-0.92\) to \(11.22\).
Mean taste preference between preferring it better *with* dip by up to \(11.2\) mm on the \(100\) mm visual analogue scale, or preferring it *without* dip by a little (up to \(-0.9\) mm on the \(100\) mm visual analogue scale.

**Ex. 26.7.**
No answer (yet).

**Ex. 26.9.**
**1.** *Diffs* are *during* minus *before*: *positive* diffs means *during* value is higher.
**2.** \(\text{s.e.}(\bar{d}) = 3.515018\).
**3.** \(-4.35\) to \(9.71\) mins.
'In the population, the mean difference between the amount of vigorous PA by Spanish health students is between \(4.35\) mins more *during* lockdown, and \(9.71\) mins more *before* lockdown.'

### Chap. 27: CIs for two means

**Ex. 27.1.**
**1.** *Diff.*: mean length of males (M) *minus* females (F).
**2.** \(\mu_M - \mu_F\).
\(\bar{x}_M - \bar{x}_F = -0.06\) m.
**3.** Plot not shown.
**4.** \(-0.25\) m to \(0.12\) m.
'The population mean difference between the length of female and male gray whales at birth has a \(95\)% chance of being between \(0.12\) m longer for male whales to \(0.25\) m longer for female whales.'
**5.** Stat. valid.

**Ex. 27.3.**
**1.** Placebo: \(0.2728678\) days; echinacea: \(0.2446822\) days.
**2.** \(0.3665054\).
**3.** Plot not shown.
**4.** \(-0.204\) to \(1.264\) days.
**5.** Placebo *minus* echinacea: the diff. between the means show how much *longer* symptoms last with placebo, compared to echinacea.
**6.** \(5.85\) to \(6.83\) days.
**7.** Stat. valid.
The diff. between the means is an average of \(0.53\) days; about half a day (quicker on echinacea).
**8.** Probably not practically important.

**Ex. 27.5.**
No answer (yet).

**Ex. 27.7.**
**1.** Either direction fine; here \(\mu_Y - \mu_O\): the amount by which younger women can lean further forward than older women.
**2.** Dot plot?
**3.** Table not shown.
**4.** Approx. CI: \(10.166\) to \(18.834^\circ\).
Exact CI (Row 2): \(9.10\) to \(19.90^\circ\).
Different: sample sizes not large.
**5.** Probably not stat. valid.
**6.** Based on the sample, a \(95\)% CI for the diff. between population mean one-step fall-recovery angle for healthy women is between \(9.1\) and \(19.9\) degrees *greater* for younger women than for older women (two independent samples).

**Ex. 27.9.**
No answer (yet).

**Ex. 27.11.**
No answer (yet).

### Chap. 28: CIs for odds ratios

**Ex. 28.1.**
One.

**Ex. 28.3.**
**1.** \(0.3000\)
**2.** \(0.3033\).
**3.** \(-0.0033\).
**4.** \(\text{s.e.}(\hat{p}_{11}) = 0.06481\); \(\text{se}(\hat{p}_{15}) = 0.04162\); \(\text{s.e.}(\hat{p}_{11} - \hat{p}_{15}) = 0.077019\).
**5.** \(-0.1573\) to \(0.1508\), larger in 2015.
**6.** \(-0.1542\) to \(0.1477\), larger in 2015.
**7.** A \(95\)% chance that the interval \(-0.1542\) to \(0.1477\) straddles the population difference in proportions.
**8.** \(0.429\).
**9.** \(0.435\).
**10.** \(0.985\).
**11.** Odds of crash involving pedestrians in 2015 \(0.985\) times as large as in 2011.
**12.** \(0.4804\) to \(.2.018\).
13. Table not shown.
14. Graph not shown.
15. Yes.

**Ex. 28.5.**
**1.** The individual standard errors: \(0.04191\) and \(0.04019\); for difference: \(0.05806\)
**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.** Not answer (yet).
**5.** \(0.3054\) to \(0.7831\) (greater for beach swimmers).
**6.** No answer (yet).
**7.** Yes.

**Ex. 28.7.**

**1.** CI: \(0.0003\) to \(0.2849\).
**2.** OR: \(2.65\); CI: \(0.979\) to \(7.174\).
**3.** No answers (yet).
**4.** Table not shown.

**Ex. 28.9.**
**1.** Table not shown.
**2.**Graph not shown.
**3.** \(0.4430\) (higher for those keeping pet birds); \(95\)% CI: \(1.605\) to \(3.174\).
**4.** \(0.1918\) (higher for those keeping pet birds); \(95\)% CI: \(0.1109\) to \(0.2727\).
**5.** Yes.

### Chap. 29: Estimating sample sizes

**Ex. 29.1.**
**1.** At least \(25\).
**2.** At least \(100\) (i.e., *four times as many*).
**3.** At least \(400\) (i.e., *sixteen times as many*.
**4.** To halve width, need *four* times as many units.
**5.** To quarter width, need *sixteen* times as many units.

**Ex. 29.3.**
**1.** At least \(10\ 000\).
**2.** At least \(2\ 500\).
**3.** At least \(1000\) needed.
**4.** Expensive (time *and* money): \(10\ 000\) and \(2\ 500\) probably unrealistic.

**Ex. 29.5.**
Use \(s = 0.43\).
**1.** At least \(1849\).
**2.** At least \(296\).
**3.** At least \(74\).
**4.** Expensive (both time *and* money); \(74\) more realistic.

**Ex. 29.7.**
Use, say, \(s = 13\).
**1.** At least \(81\) pairs.
**2.** At least \(76\) pairs.

**Ex. 29.9.**
Use, say, \(s = 0.35\).
**1.** At least \(44\) in each group.
**2.** At least \(98\) in each group.
**3.** Information not relevant to goldfish.

### Chap. 30: Tests for one proportion

**Ex. 30.1.**
**1.** One-in-five: \(0.2\).
**2.** \(H_0\): \(p = 0.2\); \(H_1\): \(p > 0.2\).
**3.** One-tailed.
**4.** Normal distribution; mean \(0.2\), std. deviation \(\text{s.e.}(\hat{p}) = 0.0444\).
**5.** \(\hat{p} = 0.6173\); \(z = 9.39\): \(P\)-value *very small*:
Very strong evidence to support the alternative hypothesis that people do better-than-guessing at identifying the placebo.

**Ex. 30.3.**
\(H_0\): \(p = 0.5\); \(H_1\): \(p \ne 0.5\).
\(\hat{p} = 0.39726\); \(\text{s.e.}(\hat{p}) = 0.05727\); \(z = -1.794\).
\(P\)-value not that small.
No evidence of difference.

**Ex. 30.5.**
\(H_0\): \(p = 0.0602\); \(H_1\): \(p < 0.602\) (*one*-tailed).
\(\hat{p} = 0.5008489\); \(n = 589\): \(\text{s.e.}(\hat{p}) = 0.0201689\), so \(z = -5.015\).
\(P\)-value very small.
Strong evidence exists that the proportion of females using the machines was lower than the proportion of females in the university population.

**Ex. 30.7.**
\(H_0\): \(p = 0.5\); \(H_1\): \(p > 0.5\) (one-tailed).
\(\hat{p} = 0.8028169\); \(n = 71\): \(\text{s.e.}(\hat{p}) = 0.05933908\), so \(z = 5.10\).
\(P\)-value very small.
Strong evidence exists that the majority of people like breadfruit pasta (for the population that the sample represents anyway).

**Ex. 30.9.**
\(H_0\): \(p = 0.15\) and \(H_1\): \(p \ne 0.15\).
\(\hat{p} = 0.06395349\) and \(n = 516\): \(\text{s.e.}(\hat{p}) = 0.01571919\), so \(z = -5.473\).
\(P\)-value very small.
Strong evidence exists that the proportion of people with CTS with a PL tendon absent is different for people with CTS.

**Ex. 30.11.**
\(H_0\): \(p = 0.5\); \(H_1\): \(p \ne 0.5\).
\(\hat{p} = 0.5576923\); \(\text{s.e.}(\hat{p}) = 0.06933752\), giving \(z = -0.8320503\).
\(P\)-value 'large'.
No evidence to suggest choice is non-random.

### Chap. 31: Tests for one mean

**Ex. 31.1.**
**1.** \(\mu\), population mean speed (in 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) \(P\)-value very small.
**7.** Very strong evidence the mean speed of vehicles on this road is greater than \(90\) km.h^{\(-1\)}.
**8.** Stat. valid.

**Ex. 31.3.**
\(H_0\): \(\mu = 50\); \(H_1\): \(\mu > 50\) (one-tailed).
\(\text{s.e.}(\bar{x}) = 4.701076\).
\(t = 7.23\): \(P\)-value very small.
*Very* strong evidence (\(P < 0.001\)) the mean mental demand is *greater* than \(50\).
(*Greater* than, because of the RQ and alternative hypothesis.)

**Ex. 31.5.**
\(H_0\): \(\mu = 14\); \(H_1\): \(\mu \ne 14\) (two-tailed).
\(\text{s.e.}(\bar{x}) = 0.09249343\).
\(t\)-score: \(10.35\), which is *huge*; \(P\)-value very small.
*Very* strong evidence (\(P < 0.001\)) that the mean weight of a Fun Size *Cherry Ripe* bar is not \(14\) g.

**Ex. 31.7.**
\(H_0\): \(\mu = 10\) (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 \(P\)-values extremely small.
For both boys and girls, very strong evidence that mean sleep time on weekend less than ten hours.

**Ex. 31.9.**
**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 the 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.** Not given.
**10.** \(n\) much larger than \(25\); stat. valid.
**11.** *Very* unlikely.

### Chap. 32: More about hypothesis tests

**Ex. 32.1.**
Use \(68\)--\(95\)--\(99.7\) rule and a 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 \(0.003\).

**Ex. 32.3.**
*Half* the values in Ex. 32.1.
**1.** Very small; certainly less than \(0.0015\) (\(99.7\)% between \(-3\) and \(3\)).
**2.** Very small; bit bigger than \(0.0015\) (\(99.7\)% between \(-3\) and \(3\)).
**3.** Bit smaller than \(0.025\) (\(95\)% between \(-2\) and \(2\)).
**4.** *Very* small; *much* smaller than \(0.0015\).

**Ex. 32.5.**
\(P\)-value *just larger* than \(0.05\); 'slight evidence' to support \(H_1\).
\(P\)-value *just smaller* than \(0.05\); 'moderate evidence' to support \(H_1\).
The difference between \(0.0501\) and \(0.0499\) is trivial though...

**Ex. 32.7.**
**1.** Hypotheses about *parameters* like \(\mu\), not *statistics* like \(\bar{x}\).
**2.** RQ two-tailed.
**3.** \(36.8052\) is a sample mean; hypothesis can be written *before* data collected.

**Ex. 32.9.**
**1.** Conclusion about the pop. **mean** energy intake.
**2.** Conclusions *never* about statistics.
**3.** The conclusion about the pop. **mean** energy intake.

**Ex. 32.11.**
Statements 2 and 4 consistent with conclusion.

### Chap. 33: Tests for paired means

**Ex. 33.1.**
How much longer the task takes on the PC for each child.

**Ex. 33.3.**
**1.** \(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d \ne 0\).
**2.** \(t = -0.205\).
**3.** \(P\) large; from software, \(P = 0.839\).
**4.** No evidence (\(t = -0.205\); two-tailed \(P = 0.839\)) of a mean increase in the weight of squash from dry to normal years (mean change: \(2.230\) g (\(95\)% CI from \(-24.8\) to \(20.3\) g), heavier in normal year).

**Ex. 33.5.**
\(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d > 0\): *diffs.* positive when dip rating better than raw.
\(t = 1.699\); *approx.* one-tailed \(P\)-value between \(16\)% and \(2.5\)%; not sure if \(P\) is larger than \(0.05\)... but likely to be (\(t\)-score quite a distance from \(z = 1\)).
The evidence *probably* doesn't support the alternative hypothesis.

**Ex. 33.7.**
\(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d > 0\); *diffs* refer to *reduction* in ferritin.
\(\bar{d} = -424.25\); \(s = 2092.693\); \(n = 20\): \(t = -0.90663\).
\(P > 0.05\) (actually \(P = 0.376\)): evidence doesn't support the alternative hypothesis.
Test may not be stat. valid; histogram of data suggests population *might* have normal distribution), though \(P\)-value is so large it probably makes little difference.

### Chap. 34: Tests for two means

**Ex. 34.1.**
How much greater the mean lymphocytes cell diameter is compared to tumour cells.

**Ex. 34.3.**
**1.** Mean length of female *minus* male.
**2.** \(H_0: \mu_F - \mu_M = 0\); \(H_1: \mu_F - \mu_M \ne 0\).
**3.** \(t = 0.65\); \(P\)-value very large.
**4.** No evidence (\(t = 0.65\); two-tailed \(P > 0.10\)) in the sample that the mean length of adult gray whales is different in the population for females (mean: \(12.70\) m; standard deviation: \(0.611\) m) and males (mean: \(12.07\) m; standard deviation: \(0.705\) m; \(95\)% CI for the difference: \(-1.26\) m to \(0.246\) m).
**5.** Statistically valid.

**Ex. 34.5.**
No answer (yet).

**Ex. 34.7.**
**1.** \(H_0\): \(\mu_I - \mu_{NI} = 0\).
\(H_1\): \(\mu_I - \mu_{NI} \ne 0\).
**2.** \(-22.54\) to \(-11.95\): mean sugar consumption between \(11.95\) and \(22.54\) kg/person/year *greater* in industrialised countries.
**3.** Very strong evidence in the sample (\(P < 0.001\)) that the mean annual sugar consumption per person is different for industrialised (mean: \(41.8\) kg/person/year) and non-industrialised (mean: \(24.6\) kg/person/year) countries (\(95\)% CI for the difference \(11.95\) to \(22.54\)).

**Ex. 34.9.**
**1.** Either direction fine; the amount by which younger (\(Y\)) women can lean further forward is \(\mu_Y - \mu_O\).
**2.** One-tailed (from RQ).
**3.** \(H_0\): \(\mu_Y - \mu_O = 0\); \(H_1\): \(\mu_Y - \mu_O > 0\).
**4.** \(t = 6.69\) (from *second row*); \(P < 0.001/2\) as one-tailed; i.e., \(P < 0.0005\).
**5.** Very strong evidence exists in the sample (\(t = 6.691\); one-tailed \(P < 0.0005\)) that the population mean one-step fall recovery angle for healthy women is *greater* for young women (mean: \(30.7^\circ\); std. dev.: \(2.58^\circ\); \(n = 10\)) compared to older women (mean: \(16.20^\circ\); std. dev.: \(4.44^\circ\); \(n = 5\); \(95\)% CI for the difference: \(9.1^\circ\) to \(19.9^\circ\)).

**Ex. 34.11.**
\(H_0\): \(\mu_M - \mu_{F} = 0\); \(H_1\): \(\mu_M - \mu_{F} \ne 0\).
From output, \(t = -2.285\); (two-tailed) \(P\)-value: \(0.024\).
Moderate evidence (\(P = 0.024\)) that the mean internal body temperature is different for females (mean: \(36.886^{\circ}\text{C}\)) and males (mean: \(36.725^{\circ}\text{C}\)).
The diff. between the means, of \(0.16\) of a degree, of little *practical* importance.

**Ex. 27.12.**
**1.** Researchers used exact \(95\)%CI; we used an approx. \(95\)% CI.
**2.** True.
**3.** False.
4. The *positive* value of \(2.76\) means coeliacs have a mean of \(2.76\) more DMFT.

So the *negative* value of \(-2.32\) means that *non*-coeliacs have a mean of \(2.32\) more DMFT.

### Chap. 35: Tests for odds ratios

**Ex. 35.1.**
Both odds: \(6.04\).

**Ex. 35.3.**
Odds: \(1.15\); Percentage: \(58.1\)%.
\(\chi^2 = 4.593\); approx. \(z = \sqrt{4.593/1} = 2.14\); expect small \(P\)-value.
Software gives \(P = 0.032\).
Stat. valid.
The sample provides *moderate evidence* (\(\text{chi-square} = 4.593\); two-tailed \(P = 0.032\)) that the *population* odds of finding a male sandfly in eastern Panama is different at \(3\) ft above ground (odds: \(1.15\)) compared to \(35\) ft above ground (odds: \(1.71\); OR: \(0.67\); \(95\)% CI from \(0.47\) to \(0.97\)).

**Ex. 35.5.**
No answer (yet).

**Ex. 35.7.**
**1.** \(6.0\)%.
**2.** \(20.5\)%.
**3.** About \(0.0640\).
**4.** About \(0.257\).
**5.** \(4.02\).
**6.** \(0.249\).
**7.** \(0.151\) to \(0.408\).
**8.** \(\chi^2 = 33.763\)% (approx. \(z = 5.81\)) and \(P < 0.001\).
**9.** Strong evidence (\(P < 0.001\); \(\chi^2 = 33.763\); \(n = 752\)) that the odds of wearing hat is different for males (odds: \(0.257\)) and females (odds: \(0.0640\); OR: \(0.249\), \(95\)% CI from \(0.151\) to \(0.408\)).
**10.** Yes.

**Ex. 35.9.**
\(\chi^2 = 22.374\), so \(z = 4.730\): very large; small \(P\)-value.
\(P < 0.001\).
The *sample* provides very strong evidence (\(\chi^2 = 22.374\); two-tailed \(P < 0.001\)) that the odds in the *population* of having a pet bird is not the same for people with lung cancer (odds: \(0.695\)) and for people without lung cancer (odds: \(0.308\); OR: \(2.26\); \(95\)% CI from \(1.6\) to \(3.2\)).

**Ex. 35.11.**
**1.** \(H_0\): No association; \(H_1\) An association.
**2.** \(23.0522\); \(P = 0.00004\).
**3.** Very strong evidence of an association.
**4.** Yes.

**Ex. 35.13.**
**1.** to **3.** Table not shown.
**4.** \(H_0\): *No association* exists between bringing a bag and age group. \(H_1\): *An association* exists between bringing a bag and age group.
**5.** \(\chi^2 = 16.24\); \(P < 0.001\).
**6.** Very strong evidence in the sample that bringing a bag is not the same for all three age groups.
**7.** Stat. valid.

### Chap. 36: Selecting an analysis

### Chap. 37: Correlation

**Ex. 37.1.**
**1.** \(H_0\): \(\rho = 0\); \(H_1\): \(\rho \ne 0\).
**2.** No evidence of (linear) relationship.
**3.** Stat. valid only if the relationship approx. linear, and variation in STAI does not change for different levels of work experience.
\(n \ge 25\).

**Ex. 37.3**.
Approx. linear; \(n = 25\); variation not constant.

**Ex. 37.5.**
**1.** Probably linear; increasing; approx. constant variance in \(y\) as \(x\) increase.
**2.** \(H_0\): \(\rho = 0\); \(H_0\): \(\rho > 0\) (one-tailed).
**3.** \(r = 0.837\); \(P < 0.001\).
Write: 'Very strong evidence exists that longer dogs also taller (\(r = 0.837\); one-tailed \(P < 0.001\); \(n = 30\))'.
**4.** Approx. linear; variation approx. constant; \(n \ge 25\): stat. valid.

**Ex. 37.7.**
**1.** \(R^2 = 0.881^2 = 77.6\)%.
About \(77.6\)% of variation in punting distance explained by variation in right-leg strength.
**2.** \(H_0\): \(\rho = 0\) and \(H_1\): \(\rho \ne 0\).
\(P\)-value *very* small; very strong evidence of correlation in population.

**Ex. 37.9.**
\(H_0\): \(\rho = 0\); \(H_1\): \(\rho \ne 0\).
\(P < 0.001\): very strong evidence of a relationship.

**Ex. 37.11.**
Non-linear relationship.

### Chap. 38: Regression

**Ex. 38.1.**
Answer very approximate.
**1.** \(r\) moderately strong, positive; \(\hat{y} = 4 + 1.7x\)
**2.** \(r\) reasonably strong, positive; \(\hat{y} = 6 + 2.3x\).
**3.** \(r\) not appropriate: variation in \(y\) increases as \(x\) increases.
**4.** \(r\) reasonably strong, negative; \(\hat{y} = 8 - 1.5x\).

**Ex. 38.3.**
**1.** \(b_0 = 3.5\); \(b_1 = -0.14\).
**2.** \(b_0 = 2.1\); \(b_1 = -0.0047\).

**Ex. 38.5.**
**1.** \(r = 0.264\).
**2.** \(R^2 = 6.97%\); using neck circumference reduces the unknown variation by about \(7\)%.
**3.** \(\hat{y} = -24.47 + 1.36x\): \(y\) is REI; \(x\) is neck circumference (in cm).
**4.** For each \(1\) cm increase in neck circumference, REI increase by an average of \(1.36\).
**5.** Approx CI: from \(0.712\) to \(2.02\).
**6.** \(t = 2.09\) and \(P = 0.041\): slight evidence of a relationship.
**7.** Stat. valid.

**Ex. 38.7.**
**1.** Intercept *not* about \(110\); line 'stops' there, but intercept is predicted value of \(y\) when \(x = 0\).
Using rise-over-run, slope about \((190 - 110)/(180 - 110) = 1.14\).
**2.** \(\hat{y} = -3.69 + 1.04x\), where \(y\) is punting distance (in feet), and \(x\) is right-leg strength (in pounds).
**3.** For each extra pound of leg strength, the punting distance increases, on average, by about \(1\) ft.
**4.** \(H_0\): \(\beta = 0\); \(H_1\): \(\beta \ne 0\).
(Could answer in terms of correlations.)
RQ two-tailed, but testing if stronger legs *increase* kicking distance seems sensible.
**5.** \(t = 6.16\): huge; \(P = 0.0001\) (two-tailed).
**6.** \(0.70\) to \(1.4\) ft.
**7.** Very strong evidence in the sample (\(t = 6.16\); \(P = 0.0001\) (two-tailed)) that punting distance is related to leg strength (slope: \(1.0427\); \(n = 13\)).

**Ex. 38.9.**
**1.** \(\hat{y} = 17.47 - 2.59x\): \(x\) is the percentage bitumen by weight; \(y\) is the percentage air voids by volume.
**2.** *Slope*: an increase in the bitumen weight by one percentage point *decreases* the average percentage air voids by volume by \(2.59\) percentage points.
*Intercept*: dodgy (extrapolation); in principle \(0\)% bitumen content by weight, the percentage air voids by volume is about \(17.47\)%.
**3.** \(t = -74.9\): massive; extremely strong evidence (\(P < 0.001\)) of a relationship.
**4.** \(\hat{y} = 4.5027\), or about \(4.5\)%.
Expected good prediction, as relationship strong.
**5.** \(\hat{y} = 1.909\), or about \(1.9\)%.
Might be a poor prediction, since extrapolation.

**Ex. 38.11.**
No answer (yet).

**Ex. 38.13.**
**1.** Too many decimal places: implies predicting \(0.0001\) of gram.
**2.** No.
**3.** Possibly; no idea of accuracy of predictions really.
**4.** *Intercept*: Weight of infant with chest circumference zero; silly.
*Slope*: average increase in birth weight (in g) for each \(1\) cm increase in chest circumference.
**5.** Intercept: cm; slope: cm/gram.
**6.** \(\hat{y} = 2\ 538.7\)g.

### Chap. 39: Writing research

**Ex. 39.1.**
**1.** to.
**2.** its.
**3.** One sample with \(50\) individuals; use 'mean' or 'median', as appropriate, not 'average'.
**4.** Should be one sentence.

**Ex. 39.3.**
**1.** Ambiguous; sound like cage is male; passive voice.
'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.'

**Ex. 39.5.**
Number decimal places ridiculous.

**Ex. 39.7.**
RQ: P, O, C and I unclear; fonts should be identified.
Perhaps better: For students, is the mean reading speed for text in the Georgia font the same as for text in Calibri font?
**Abstract** statement poor (*fonts* are not fast or slow).
Perhaps: The sample provided evidence that the mean reading speeds were different (\(P = ???\)), when comparing text in Georgia font (mean: ???) and Calibri font (mean: ???; \(95\)% CI for the difference: ??? to ???).

**Ex. 39.9.**
Variables *qualitative*: means inappropriate; use odds ratio; values almost certainly refer to the CI for the OR.
Without more information, we can't be sure what the OR means.

### Chap. 40: Reading research

**Ex. 40.1.**
**1.** Convenience; self-selected.
Those in the study *may* record different accuracies than people not in the study.
**2.** Inclusion criteria.
**3.** Ethical (drop-outs happen); accurate description of study.
**4.** Not ecologically valid.
**5.** Paired \(t\)-test.
**6.** Null: No mean difference between counts on phone and manually counted; alternative: a difference.
**7.** \(P\) small; evidence that the mean difference in step-count between the two methods cannot be explained by chance: likely is a difference.
**8.** Valid.

**Ex. 40.3.**
**1.** Only some evidence of diff. in mean age.
**2.** Comparing the two groups; age a possible confounder.
**3.** Two-sample \(t\)-test.
**4.** \(0.03376\).
**5.** \(t = 2.07\); small \(P\)-value.
**6.** Evidence of difference.
**7.** Probably, as given standard errors rounded.
**8.** Conceptual.
**9.** Table not shown.
**10.** \(\chi^2\).
**11.** \(z = 1.75\); \(P\) between \(5\)% and \(32\)%: not helpful.
**12.** Observational, so not cause-and-effect; no confounders noted; very restricted population.

**Ex. 40.5.**
**1.** \(\chi^2\)-test to compare proportions.
**2.** No evidence of difference in survival rates at the temperatures.
**3.** Evidence that surviving *Cx.* had a 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.** For *Cx.*, evidence the mean sizes at the two temperatures were different; for *Ae.*, no evidence the mean sizes at the two temperatures were different.
**10.** Two-sample \(t\)-tests.
**11.** Intercept: \(-55.40\) to \(16.28\); slope: \(3.88\) to \(59.40\).
**12.** \(t = 2.28\); expect small \(P\)-value; evidence of linear association.
**13.** Need scatterplot to be sure, but \(n \ge 25\).
**14.** When the predator--size ratio increases by one, predation efficiency increases by \(31.64\) percentage points.
**15.** About \(8.7\)% of variation in predation efficient can be explained by the value of the predator--size ratio.
**16.** \(r = 0.294\).

**Ex. 40.7.**
**1.** Stratified?
**2.** Two-sample \(t\)-test.
**3.** Strong evidence the mean number of actinomycetes are different.
**4.** Possibly not; sample sizes small (does not mean results useless!).
**5.** Two-sample \(t\)-test.
**6.** Very strong evidence that mean number higher in CNV farms.
**7.** Larger actinomycetes numbers linearly associated with lower corky root severity.
**8.** \(R^2 = 57.8\)%; \(57.8\)% of variation in corky root severity explained by actinomycete abundance.