# E Answers to end-of-chapter exercises

### Chap. 2: RQs

**Ex. 2.1.**
**1.** *Percentage* of vehicles that crash.
**2.** *Average* height people can jump.
**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, online? undergraduate, postgraduate? full-time, part-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
(levels: 'Talking on the phone'; 'texting on the phone').
**5.** Walking speed.
**6.** *Average* walking speed.

**Ex. 2.13**:
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. 2.15**:
**1.** Japanese adults.
**2.** Between those who take and do not take Vitamin C tablets.
**3.** *Average* cold duration'.
**4.** Duration of cold symptoms for each person.
**5.** Whether or not each person takes Vitamin C tablets or not.
**6.** Decision-making
**7.** One-tailed.

**Ex. 2.17**:
**1.** Units of analysis: person; unit of 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. 2.20.**
**1.** Descriptive; estimation.
**2.** Descriptive; decision-making.
**3.** Correlational; estimation.
**4.** Relational; decision-making.

Unclear if intervention; seems unlikely.

**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 cars do not operate independently: 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. 2.24.**
Analysis: \(12\) subjects.
Observation: \(6\) per subjects: \(72\).

### Chap. 3: Types of study designs

**Ex. 3.1.**
**1.** Between-individuals.
**2.** Relational.
**3.** Most likely.
**4.** Estimation.
**5.** Intervention, so experiment.
Likely true experiment.

**Ex. 3.3**:
*True experiment*.

**Ex. 3.5**:
*Quasi-experiment*.

**Ex. 3.7**:
**1.** Diet.
**2.** Change in body weight after \(2\) years.
**3.** Experimental: diets *manipulated* and *imposed* by the 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 two years.
**8.** Type of diet.

### Chap. 5: Sampling

**Ex. 5.1.**
c. Externally-valid study more likely.

**Ex. 5.2.**
d. Precise estimates more likely.

**Ex. 5.3.**
**1.** Every \(7\)th day is same day of week.
**2.** Maybe select days at random over three-month period.

**Ex. 5.5**:
**1.** Multi-stage.
**2.** Stratified (selecting floor), then convenience.
**3.** Convenience.
**4.** Part stratified (selecting floors), then convenience.
First might be best.

**Ex. 5.7**:
Random sampling to select schools.
Then, self-selecting.

**Ex. 5.9**:
Stratified: zones are strata.

**Ex. 5.10.**
No answer (yet).

**Ex. 5.11.**
In Stage 2, selection of farms *not* random.

**Ex. 5.12.**
In Stage 1, selection of schools *not* random.

### Chap. 6: Overview of internal validity

**Ex. 6.1.**
**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. 6.3**:
*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).

### Chap. 7: Designing experimental studies

**Ex. 7.1.**
Only Statement 5 is true.

**Ex. 7.3.**
Lurking; confounding.

**Ex. 7.5.**
**1.** A group receiving a pill like Treatment A and B, with no effective ingredient.
**2.** Blinding participants.
**3.** To ensure participants do not change behaviour because of the treatment received.

**Ex. 7.7**:
Observer effect.
Researcher directly contacting the subjects; may unintentionally influence responses.

**Ex. 7.9.**
**1.** Random allocation; blocking; recording potential confounders.
**2.** Blinding particiipants, researchers.
**3.** Change in nasal congestion.
**4.** Type of cleaning.
**5.** Age; sex.

### Chap. 8: Designing observational studies

**Ex. 8.1.**
Second statement is true.

**Ex. 8.3**:
*Random allocation*: not possible (observational).
*Blinding*: students unaware of which water they drink; in observational study, probably infeasible.
*Double blinding*: neither students nor researchers know which type of water students are drinking; probably infeasible.
*Control*: not sensible.
*Random sample*: any random sampling method preferred; possible, but unlikely.

**Ex. 8.5**:
Patients probably knew they were involved; Hawthorne effect should be considered in interpretation.

**Ex. 8.7**:
**1.** *Response*: amount of sunscreen used; *explanatory*: time applying sunscreen.
**2.** Potential confounding variables.
**3.** If the mean of both the response and explanatory variables was different for females and males, sex of the participant would be a *confounder*; would need to be factored into the data analysis.
**4.** Participants blinded to what is happening in study.

### Chap. 9: Research design limitations

**Ex. 9.1.**
External.

**Ex. 9.3**:
Population: 'on-campus university students where I work'.
External validity: whether the results apply to other members of *target population*.

**Ex. 9.5**:
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. 9.7.**
Observational study: having a severe cough mean people take more cough drops.

### Chap. 10: Collecting data

**Ex. 10.1**:
No place for \(18\)-year-olds.

**Ex. 10.3**:
Best: second.
First: *leading* (*concerned* cat owners...)
Third: *leading* (Do you *agree*...)

**Ex. 10.5**.
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. 11: Classifying data

**Ex. 11.1.**
Quant. continuous.
Qual. nominal.
Quant. continuous.

**Ex. 11.3.**
False; true; false

**Ex. 11.5.**
Nominal; qualitative.

**Ex. 11.7**:
**1.** Blood pressure: quant. continuous.
**2.** Program: qual. nominal.
**3.** Grade: qual. ordinal.
**4.** Number of doctor visits: quant. discrete.

**Ex. 11.9**:
*Gender*: Qual. nominal.
*Age*: Quant. continuous.
*Height*: Quant. continuous.
*Weight*: Quant. continuous.
*GMFCS*: Qual. ordinal.

**Ex. 11.11**:
*Kangaroo response*: qual. ordinal (perhaps nominal?).
*Drone height*: quant.; with just four values used, probably treated as qual. ordinal.
*Mob size*: quant. discrete.
*Sex*: qual. nominal.

### Chap. 12: Summarising quantitative data

**Ex. 12.1.**
Average: perhaps \(70\)--\(80\)?
Variation: most between \(30\) and \(80\).
Shape: skewed *left*.
Outliers: none; 'bump' at lower ages.

**Ex. 12.2.**
Very right-skewed.
Average is somewhere near 1 or 2 k perhaps.
Most are between 0 and about 5 kg.
Possible outlier near 11 kg.

**Ex. 12.3**:
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. 12.4.**
Average around \(15.0\) g.
Most between \(13.5\) and \(16.0\) kg.
Slightly skewed left, with perhaps one low outlier.
Maybe quote *both* the mean and median (\(14.90\) and \(14.99\) respectively).

**Ex. 12.5**:
**1.** \(3.7\).
**2.** \(3.5\).
**3.** \(1.888562\).

**Ex. 12.7**:
**1.** Mean: \(0.467\).
**2.** Median: \(3.35\).
**3.** Range: \(29.6\) (from \(-19.8\) to \(9.8\)).
**4.** Std dev.: \(10.40263\).
(No units of measurement.)

**Ex. 12.9.**
**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.** No answer.
**3.** No answer.
**4.** No answer.
**5.** Hard to describe with standard language.

**Ex. 12.10.**
*Average*: hard to be sure... maybe between \(10\) or \(15\).
*Variation*: about \(0\) to about \(40\).
*Shape*: slightly skewed right.
*Outliers*: no outliers or unusual observations; the observation between \(35\) and \(40\) *may* be an outlier.
I suspect it is *not* an outlier, as a larger sample may very well have observations between \(30\) and \(35\).
Of course, I could be wrong.

**Ex. 12.12.**
In Dataset B, more observations are close to the mean; the average distance would be a small number.
The standard deviation for Dataset B will be **smaller** than the standard deviation for Dataset A.

### Chap. 13: Summarising qualitative data

**Ex. 13.1.**
Most common social group: many females plus offspring.
No commonly-observed social group include males.
Graph not shown.

**Ex. 13.3.**
*Age*: histogram.
*FEV*: histogram.
*Height*: histogram.
*Gender*: bar or dot; mode: male (\(51.4\)%; odds: \(1.06\)).
*Smoking*: bar or pie; mode: non-smoking (\(9.9\)%; odds: \(0.11\)).

**Ex. 13.5**:
Bar (or dot) chart.
Pie chart inappropriate: more than one option can be selected.

**Ex. 13.7.**
Plots not shown.
**4.** Advantage: availability. Disadvantage: high price.
**5.** Table not shown.
**6.** \(40/(231 - 40) = 0.209\).
**7.** \(40/(12 + 21) = 1.21\).

### Chap. 14: Qualitative data: Comparing between individuals

**Ex. 14.1**:
**1.** *Vomited*: \(0.50\) beer then wine; \(0.50\) wine only.
*Didn't vomit*: \(0.738\) beer then wine, \(0.262\) wine only.
Proportion 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.
The 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.** \(0.354\).

**Ex. 14.3**:
**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 of no August rainfall in Emerald \(2.7\) times higher in months with non-positive SOI.

**Ex. 14.5.**
Plot not shown.

**Ex. 14.6.**
**1.** *Proportion* F skipped: \(\hat{p}_F = 0.359\);
**2.** *Proportion* M skipped: \(\hat{p}_M = 0.284\).
**3.** \(\text{Odds}(\text{Skips breakfast, F}) = 0.5598\);
**4.** \(\text{Odds}(\text{Skips breakfast, M}) = 0.3966\).
**5.** *Odds ratio*: \(1.41\).
**6.** Odds of females skipping are \(1.41\) *times* the odds of males skipping
**7.** Not shown.

**Ex. 14.7.**
**1.** \(13.2\)%.
**2.** \(2.3\)%.
**3.** \(0.152\).
**4.** \(0.0238\).
**5.** \(6.39\).
**6.** Odds of coffee drinker being a smoker is \(6.39\) times the odds of a non-coffee dribker being a smoker.
**7.** Not shown.

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

### Chap. 16: Quantitative data: Comparing between individuals

**Ex. 16.1.**
The DB method, in general, produces smaller cost over-runs.

**Ex. 16.3**:
**A**: II (median; IQR).
**B**: I (mean; standard deviation).
**C**: III (median; IQR).

**Ex. 16.5**:
Plot not shown.

**Ex. 16.7.**
No answer (yet).

**Ex. 16.9.**
**1.** Table not shown.
**2.** Plot not shown.

**Ex. 16.11.**
**1.** `mAcc`

: highly *left* skewed; `Age`

: highly *right* skewed; `mTS`

: slightly right skewed.
Perhaps use medians, IQRs for summarising (mean, std dev. probably OK for `mTS`

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

### Chap. 17: Quantitative data: Correlations between individuals

**Ex. 17.1**:
Many correct answers.

**Ex. 17.1**:
You cannot be very precise with answers.
A: Large; positive.
B: Moderate; negative.
C: Close to zero.
D: Not appropriate.

**Ex. 17.3.**
You cannot be very accurate!
From software: \(r = 0.71\).
The best you can do is 'a reasonably high positive \(r\) value'.

**Ex. 17.5.**
**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. 17.7**:
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. 17.9**:
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. 17.11.**
No answer (yet).

### Chap. 18: More about summarising data

**Ex. 18.1.**
Scatterplot; histogram of the diffs; side-by-side bar.

**Ex. 18.2**:
Individual variables: *bar chart* for origin; *histogram* for others.
*Relationships* are main focus.
Between biomass, origin: *boxplot*.
Between biomass, other variables: *scatterplot*.
(On scatterplot, origins could be encoded with different colours or symbols.)

**Ex. 18.4**:
*Fertilizer* (quant.): histogram (response).
*Soil nitrogen* (quant.): Histogram (explanatory).
*Source* (qual. nominal): Bar chart (explanatory).
*Relationships*: Between fertilizer dose, soil nitrogen: scatterplot.
*Source* could be encoded using different coloured points.

**Ex. 18.5**:
Plotting symbols unexplained.
Axis labels unhelpful.
Vertical axis could stop at \(20\).

**Ex. 18.6**:
Graph *inappropriate*: both variables *qualitative*.
Use stacked or side-by-side bar chart.

**Ex. 18.7**:
**1.** Response: *change* in MADRS (quant. continuous).
**2.** Explanatory: treatment group (qual. nominal, three levels).
**3.** Response: histogram. Explanatory: bar chart. Relationship: boxplot.

**Ex. 18.8**:
Variable is 'Sport' (*qual. nominal*).
The bars can be ordered any way.
*Skewness makes no sense*: only sensible for *quant.* variables.

**Ex. 18.9.**
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?

**Ex. 18.10**:
**Graph**: odd colour choice; vertical axis label unhelpful; horizontal axis isn't unlabelled; units of measurement not given; title and/or caption helpful.

**Table**: CI limits under the *Mean* and *Std dev* columns; units of measurement not given; no caption, or explanation of table; number of decimal places is inconsistent; sample sizes not given; *difference* (and prob. other rows) should report a std error.

### Chap. 19: Probability

**Ex. 19.1.**
**1.** Subjective.
**2.** Rel. frequency.

**Ex. 19.3.**
False; true; depends on first card.

**Ex. 19.5.**
\(3/6\); \(5\); yes: what happens on die won't change coin outcome.

**Ex. 19.7**:
No answer (yet).

**Ex. 19.9**:
**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. 19.11**:
Events not equally likely.

**Ex. 19.12.**
**1.** \(49.2\)%.
**2.** \(17.7\)%.
**3.** \(2383/4327 = 55.1\)%.

### Chap. 20: Making decisions

### Chap. 21: Sampling variation

**Ex. 21.1**:
**1.** Std dev.
**2.** Std error (of mean).
**3.** Std dev.
**4.** Std error (of proportion).

**Ex. 21.2**:
**1.** No: \(p\) doesn't vary from sample to sample.
**2.** Yes: varies from sample to sample.
**3.** Yes: varies from sample to sample.
**4.** Yes: varies from sample to sample.
**5.** No: *Population* odds don't vary from sample to sample.

**Ex. 21.3**:
The *standard error of the mean* describes how the sample mean varies from sample to sample.
Describes precision of \(\bar{x}\) for estimating \(\mu\)

### Chap. 22: Distributions and models

**Ex. 22.1.**
*All* are false.

**Ex. 22.3**:
**1:** C; **2:** A; **3:** B; **4:** D.

**Ex. 22.5**:
**1.** \(z = -0.30\); about \(38.2\)%.
**2.** \(z = 0.07\); about \(47.2\)%.
**3.** The \(z\)-scores: \(-0.67\) and \(0.44\); \(0.6700 - 0.2514\); about \(41.9\)%. (Diagram!)
**4.** \(z\)-score about \(1.04\); tree diameter \(x = 8.8 + (1.04\times 2.7)\); about \(11.6\) inches.

**Ex. 22.7**:
**1.** \(z = -0.61\); \(1 - 0.2709\), or about \(72.9\)%.
**2.** \(z = -1.83\); about \(3.4\)%.
**3.** The \(z\)-scores: \(-4.878\) and \(-1.83\); about \(3.4\)%.
**4.** The \(z\)-score: \(1.64\) (or \(1.65\)).
Gestation length: \(x = 42.7\).
\(5\)% of gestation lengths *longer* than about \(42.7\) weeks.
**5.** \(z\)-score: \(-1.64\) (or \(-1.65\)).
Gestation length: \(x = 40 + (-1.64 \times 1.64)\); \(5\)% of gestation lengths *shorter* than about \(37.3\) weeks.

**Ex. 22.9**:
Lower than about \(80.8\): rejection.

### Chap. 24: CIs for one proportion

**Ex. 24.1**:
\(\hat{p} = 0.8194444\) and \(n = 864\).
\(\text{s.e.}(\hat{p}) = 0.01309\); approx. \(95\)% CI: \(0.819 \pm (2\times 0.0131)\).
Statistically valid.

**Ex. 24.3**:
\(\hat{p} = 0.05194805\); \(\text{s.e.}(\hat{p}) = 0.0017833\); approx. \(95\)% CI: \(0.0519\pm 0.0358\).
Statistically valid.

**Ex. 24.5**:
\(\hat{p} = 0.317059\); \(n = 6882\).
\(\text{s.e.}(\hat{p}) = 0.005609244\).
CI: \(0.317\pm 0.011\).
Statistically valid.

**Ex. 24.7.**
\(\hat{p} = 365/1516 = 0.241\).
\(\text{s.e.}(\hat{p}) = 0.01098449\).
Approx. \(95\)% CI: \(0.219\) to \(0.263\).

### Chap. 25: CIs for one mean

**Ex. 25.1.**
**1.** *Parameter*: population mean weight of an American black bear, \(\mu\).
**2.** \(\text{s.e.}(\bar{x}) = \frac{s}{\sqrt{n}} = 51.1/\sqrt{185} = 3.756947\).
**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.** Statistically valid as \(n > 25\).

**Ex. 25.3**:
\(\text{s.e.} = 0.06410062\).
*Approx.* \(95\)% CI: \(2.72\) L to \(2.98\) L.

**Ex. 25.5**:
Approx. \(95\)% CI : \(29.9\) s to \(36.1\) s.

**Ex. 25.7**:
*None* acceptable.
**1.** CIs not about observations, but *statistics* (nor parameters).
**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. 25.9**:
\(\text{s.e.}(\bar{x}) = 5.36768\); approx. \(95\)% CI: \(50.56\) s to \(72.04\) s.
Statistically valid.

### Chap. 26: More about CIs

### Chap. 27: CIs for paired data

**Ex. 27.1.**
**1.** Paired.
**2.** Paired.

**Ex. 27.3.**
**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. 27.5**:
Mean of *differences*: \(5.2\); 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. 27.7.**
No answer (yet).

**Ex. 27.9.**
**1.** *Differences* are *during* minus *before*: *positive* differences 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. 28: CIs for two means

**Ex. 28.1.**
**1.** *Difference*: mean length of females *minus* the mean length of males.
(Either direction is fine.)
**2.** \(\mu_F - \mu_M\) (\(F\) and \(M\) represent female, male gray whales respectively).
Estimate: \(\bar{x}_F - \bar{x}_M = 4.66 - 4.60 = 0.06\) m.
**3.** Plot not shown.
**4.** \(-0.12\) m to \(0.25\) 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.** Both sample sizes are larger than \(25\): statistically valid.

**Ex. 28.3**:
**1.** Placebo: \(0.2728678\) days; echinacea: \(0.2446822\) days.
**2.** \(-0.204\) to \(1.264\) days.
**3.** Placebo *minus* echinacea: the diff. between the means show how much *longer* symptoms last with placebo, compared to echinacea.
**4.** \(5.85\) to \(6.83\) days.
**5.** Sample sizes large; statistically valid.
The difference between the means is an average of \(0.53\) days; about half a day (quicker on echinacea).
Probably not practically important.

**Ex. 28.4**:
**1.** Exercise group: \(0.4427189\); splinting: \(0.3478505\).
**2.** Splinting *minus* exercise: the difference are how much greater the pain is with splinting.
**3.** \(-0.826\) to \(1.426\): \(0.826\) greater pain with exercise to \(1.426\) greater pain with splinting.
**4.** \(0.404\) to \(1.796\).
**5.** Sample sizes are small; CIs may not be statistically valid, roughly correct only.

**Ex. 28.6**
**1.** Perhaps: The \(\mu_{\text{After}} - \mu_{\text{Before}}\), the *increase* in deceleration.
**2.** Approx. CI: \(-0.00162\) to \(0.00562\) m/s.
The difference between the mean decelerations is likely to be somewhere between \(-0.0016\) m/s (i.e, a mean acceleration of \(0.0016\) m/s) to \(0.0056\) m/s.

**Ex. 28.7**
**1.** Either direction fine; we use \(\mu_Y - \mu_O\): the amount by which younger women can lean further forward than older women.
**2.** Small dataset... dot plot?
**3.** Table mot shown.
**4.** Approx. CI: (No answer yet).
Exact CI: From Row 2: \(9.10\) to \(19.90\).
Different, as sample sizes not large.
**5.** Both sample sizes are less than \(25\); probably not statistically 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).

### Chap. 29: CIs for odds ratios

**Ex. 29.1.**
**1.** Table not shown.
**2.** \(15\div 35 = 0.429\).
**3.** \(37\div 85 = 0.435\).
**4.** \(0.429\div 0.435 = 0.985\).
**5.** \(0.480\) to \(2.018\).
**6.** The population OR for a crash involving pedestrians, comparing 2011 (odds: \(0.429\); \(n = 50\)) to 2015 (odds: \(0.435\); \(n = 122\)) is \(0.986\), and has a \(95\)% chance of being between \(0.480\) and \(2.018\).
**7.** All *expected* counts larger than five; statistically valid:

**Ex. 29.3**:
Odds of swimming at the beach; OR compares these odds between those without to those with an ear infection.
*Or*: odds of *not* having an ear infection; OR compares these odds for beach to non-beach swimmers.

**Ex. 29.5**:
Odds no rainfall (non-pos. SOI): \(14/40 = 0.35\).
Odds no rainfall (neg. SOI): \(7/53 = 0.1320755\).
OR: \(0.35/0.1320755 = 2.65\), as in output.
\(95\)% CI: \(0.979\) to \(7.174\).

**Ex. 29.7**:
Numerical summary: not shown.
Graphical summary: not shwon
*Sample* OR: \(2.257\); \(95\)% CI: \(1.605\) to \(3.174\).
Based on the sample, a \(95\)% CI for the OR of keeping a pet bird is from \(1.605\) to \(3.174\) (comparing people *with* lung cancer to those *without* lung cancer).
The CI statistically valid: all expected counts exceed five.

### Chap. 30: Estimating sample sizes

**Ex. 30.1**:
**1.** \(n = 1/0.04^2 = 625\); at least \(625\).
**2.** \(n = 1/0.02^2 = 2500\); at least \(2500\) (i.e., *four times as many*).
**3.** \(n = 1/0.01^2 = 10,00\); at least \(10\ 000\) (i.e., *sixteen times as many*.
**4.** To halve the width, need *four* times as many units.
**5.** To quarter the width, need *sixteen* times as many units.

**Ex. 30.3**:
Use \(s = 0.43\).
**1.** \(n = \left( (2\times 0.43)/0.02 \right)^2 = 1849\); at least \(1849\).
**2.** \(n = 295.85\); at least \(296\).
**3.** \(n = 73.96\); at least \(74\).
**4.** Expensive (both time *and* money); \(74\) more realistic.

**Ex. 30.5**:
**1.** Approx. \(n = 1/(0.05^2) = 400\).
**2.** Approx. \(n = 1/(0.025^2) = 1600\).
**3.** To *halve* the width, *four* times as many people needed.

### Chap. 31: Tests for one proportion

**Ex. 31.1.**
**1.** One-in-five: \(0.2\).
**2.** \(H_0\): \(p = 0.2\); \(H_1\): \(p > 0.2\).
**3.** One-tailed.
**4.** \(\text{s.e.}(\hat{p}) = 0.044444\); normal distribution centred on \(0.2\), std deviation \(\text{s.e.}(\hat{p}) = 0.0444\).
**5.** \(\hat{p} = 0.6173\); \(z = 9.39\): \(P\)-value will be *very small*:
Very strong evidence to support the alternative hypothesis that people do better-than-guessing at identifying the placebo.

**Ex. 31.3.**
No answer (yet).

**Ex. 31.5.**
\(H_0\): \(p = 0.0602\) and \(H_1\): \(p < 0.602\) (*one*-tailed).
\(\hat{p} = 0.5008489\) and \(n = 589\): \(\text{s.e.}(\hat{p}) = 0.0201689\), so \(z = -5.015\).
\(P\)-value will be 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. 31.7.**
\(H_0\): \(p = 0.5\) and \(H_1\): \(p > 0.5\) (one-tailed).
\(\hat{p} = 0.8028169\) and \(n = 71\): \(\text{s.e.}(\hat{p}) = 0.05933908\), so \(z = 5.10\).
\(P\)-value will be very small.
Strong evidence exists that the majority of people like breadfruit pasta (for the population that the sample represents anyway).

**Ex. 31.9.**
\(H_0\): \(p = 0.15\) and \(H_1\): \(p \ne 0.15\).
\(\hat{p} = 33/516 = 0.06395349\) and \(n = 516\): \(\text{s.e.}(\hat{p}) = 0.01571919\), so \(z = -5.473\).
\(P\)-value will be very small.
Strong evidence exists that the proportion of people with CTS with a PL tendon absent is different for people with CTS.

**Ex. 31.11.**
\(H_0\): \(p = 0.5\) and \(H_1\): \(p \ne 0.5\).
\(n = 52\), so \(\hat{p} = 0.5576923\): \(\text{s.e.}(\hat{p}) = 0.06933752\), giving \(z = -0.8320503\).
\(P\)-value will be 'large'.
No evidence to suggest that choice is non-random.

### Chap. 32: Tests for one mean

**Ex. 32.1.**
**1.** \(\mu\), the population mean speed (in km.h^{-1}).
**2.** \(\text{$H_0$: } \mu = 90\) and \(\text{$H_1$: }\mu > 90\) (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 (\(t = 9.46\); one-tailed \(P < 0.001\)) that the mean speed of vehicles on this road (sample mean: \(96.56\) (approx. \(95\)% CI: \(95.17\) to \(97.95\)); standard deviation: \(13.874\)) is greater than \(90\) km.h^{-1}.

**Ex. 32.3**:
\(H_0\): \(\mu = 50\) and \(H_1\): \(\mu > 50\) (one-tailed); \(\mu\) is mean mental demand.
\(\text{s.e.}(\bar{x}) = 4.701076\).
\(t = 7.23\): \(P\)-value will be very small.
*Very* strong evidence (\(P < 0.001\)) that the mean mental demand is *greater* than \(50\).
(Notice we say *greater* than, because of the RQ and alternative hypothesis.)

**Ex. 32.5.**
\(H_0\): \(\mu = 10\) (or \(\mu \ge 10\)) and \(H_1\): \(\mu < 10\).
*Females*: \(\text{s.e.}(\bar{x}) = 0.05924742\); \(t = -25.32\): \(P\)-value extremely small.
*Males*: \(\text{s.e.}(\bar{x}) = 0.0700152\); \(t = -19.42\): \(P\)-value extremely small.
For both boys and girls, very strong evidence exists (girls: \(t = -25.32\); boys: \(t = -19.42\); \(P < 0.001\) for both) that the mean sleep time one weekend is less than ten hours (girls: mean \(8.64\) hrs; boys: \(8.50\) hrs).

**Ex. 32.7**:
\(H_0\): \(\mu = 1000\) and \(H_1\): \(\mu \ne 1000\) (\(\mu\) is the population mean guess of spill volume).
Std error: \(46.15526\).
\(t = -3.33\): \(P\)-value very small.
Very strong evidence that the mean guess of blood volume is not \(1000\) ml, the actual value.
The sample is much larger than \(25\): the test is statistically valid.

### Chap. 33: More about hypothesis tests

**Ex. 33.1**:
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. 33.3**:
*Half* the values in **Ex. 33.1**.
**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. 33.5**:
\(P\)-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. 33.7**:
**1.** Hypotheses are about *parameters* like \(\mu\), not *statistics* like \(\bar{x}\).
**2.** Fine if one-tailed RQ.
**3.** \(36.8052\) is a sample mean; hypothesis can be written down *before* data are collected.

**Ex. 33.9**:
**1.** Conclusion about the population **mean** energy intake.
**2.** Conclusions *never* about statistics.
**3.** The conclusion about the population **mean** energy intake.

### Chap. 34: Tests for paired means

**Ex. 34.1.**
**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. 34.3**:
\(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d > 0\): *differences* positive when dip rating better than raw rating.
\(t = 1.699\); *approx.* one-tailed \(P\)-value (using the \(68\)--\(95\)--\(99.7\) rule) between \(16\)% and \(2.5\)%.
So we cannot be sure if the \(P\)-value is larger than \(0.05\)... but it is likely to be (the calculated \(t\)-score is quite a distance from \(z = 1\)).
The evidence *probably* doesn't support the alternative hypothesis.

**Ex. 34.5**:
\(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d > 0\); *differences* refer to *reduction* in ferritin.
\(\bar{d} = -424.25\); \(s = 2092.693\); \(n = 20\): \(t = -0.90663\).
\(P > 0.05\) (actually \(P = 0.376\)): the evidence doesn't support the alternative hypothesis.
Since \(n < 25\), test may not be statistically valid (the histogram of data suggests population *might* have a normal distribution), though \(P\)-value is so large it probably makes little difference.

### Chap. 35: Tests for two means

**Ex. 35.1.**
**1.** The mean length of female *minus* male.
**2.** \(H_0: \mu_F - \mu_M = 0\) and \(H_1: \mu_F - \mu_M \ne 0\).
**3.** \(t = 0.65\); the \(P\)-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. 35.3.**
No answer (yet).

**Ex. 35.5**:
**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. 35.7.**
**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 the *second row*); \(P < 0.001/2\) since 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. 35.9**:
\(H_0\): \(\mu_M - \mu_{F} = 0\); \(H_1\): \(\mu_M - \mu_{F} \ne 0\).
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 difference between the means, of \(0.16\) of a degree, of little *practical* importance.

### Chap. 36: Tests for odds ratios

**Ex. 36.1**:
Odds: \(1.15\); Percentage: \(58.1\)%.
\(\chi^2 = 4.593\); approx. \(z = \sqrt{4.593/1} = 2.14\); expect small \(P\)-value.
Software gives \(P = 0.032\).
Statistically 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. 36.3.**
No answer (yet).

**Ex. 36.5**:
**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. 36.7**:
From software: \(\chi^2 = 22.374\), approx. \(z = \sqrt{22.374/1} = 4.730\): very large; small \(P\)-value.
From software: \(P < 0.001\).

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 \(3.2\)).

### Chap. 37: Selecting an analysis

### Chap. 38: Correlation

NEED NEW EXERCISE

**Ex. 38.3.**
**1.** \(H_0\): \(\rho = 0\) and \(H_1\): \(\rho \ne 0\).
**2.** No evidence of (linear) relationship.
**3.** No scatterplot provided; statistically valid only if the relationship approx. linear, and variation in STAI does not change for different levels of work experience.
The sample size larger than \(25\).

**Ex. 38.5**:
The plot looks linear; \(n = 25\); variation not constant.

**Ex. 38.7**:
**1.** \(R^2 = 0.881^2 = 77.6\)%.
About \(77.6\)% of variation in punting distance explained by the variation in right-leg strength.
**2.** \(H_0\): \(\rho = 0\) and \(H_1\): \(\rho \ne 0\).
\(P\)-value *very* small; very strong evidence of a correlation in the population.

**Ex. 38.9.**
\(H_0\): \(\rho = 0\) and \(H_1\): \(\rho \ne 0\).
\(P < 0.001\): very strong evidence of a relationship.

**Ex. 38.11.**
Non-linear relationship.

### Chap. 39: Regression

**Ex. 39.1**:
**1.** *Way* too many decimal places.
\(r\) not relevant: relationship non-linear.
**2.** Regression inappropriate: relationship non-linear.
**3.** \(y\) should be \(\hat{y}\); slope, intercept values *swapped*.
**4.** The whole thing is bothersome...

**Ex. 39.3.**
**1.** \(\hat{y} = 150.19 - 0.348x\) (\(y\): mean number of 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 \(0.35\).
Perhaps 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 \(P\)-value very small: \(P < 0.001\).

**Ex. 39.9**:
**1.** \(b_0\): *No* time spent on sunscreen application, average of \(0.27\) g has been applied; nonsense.
\(b_1\): Each extra minute spent on application adds an average of \(2.21\) g of sunscreen: sensible.
**2.** \(\beta_0\) could be zero... which would make sense.
**3.** \(\hat{y} = 18\) g.
**4.** About \(64\)% of the variation in sunscreen amount applied can be explained by the variation in the time spent on application.
**5.** \(r = \sqrt{0.64} = 0.8\) (must be positive value).
A strong positive correlation between the variables.

### Chap. 40: Writing research

**Ex. 40.1**:
No. of decimal places ridiculous.

**Ex. 40.2**:
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. 40.3**:
No units of measurement; jump-heights given to \(0.001\) of a centimetre.; table could also summarise information for each individual jump type; numerical summary shouldn't include \(P\)-value, \(t\)-score, or CI.

**Ex. 40.4**:
Variables *qualitative*: means inappropriate; appropriate summary is odds ratio, so values almost certainly refer to the CI for the OR.
Without more information, we can't really be sure what the OR means though.

**Ex. 40.5**:
This study alone cannot *prove* anything; *difference* between hang times is of interest: appropriate CI is for difference between the mean hang times.

**Ex. 40.6**:
**1.** Table: reasonably good!
**2.** Figure: poor (3D).
Use a stacked or side-by-side bar chart.

**Ex. 40.7**:
Reasonably good: no gaps between histogram bars.

### Chap. 41: Reading research

**Ex. 41.1**:
**1.** Not ecologically valid.
**2.** Ethical. People understand that sometimes unexpected things happen.
**3.** Convenience; self-selected.
Nothing obvious suggests those in the study would record different accuracies than people not in the study.
**4.** Inclusion criteria.
**5.** Paired \(t\)-test.
**6.** Evidence in the sample that the mean difference in step-count between the two methods cannot be explained by chance: likely is a difference.
**7.** From the given information: probably valid.

**Ex. 41.3.**
No answer (yet).