# E Answers to end-of-chapter exercises

### Chap. 1: Introduction

Ex. 1.1: Quantitative.

Ex. 1.3. Quantitative.

### 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. 5.11. 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

Ex. 15.1. 1. House. Graphs not shown.

### 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. 16.9. 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: 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.

### 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. 19.9: 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

Ex. 20.1: 1. Yes! Problem seems likely (we can't be sure). 2. Assuming fair die, would not expect a ten times in a row.

Ex. 20.3. Seems unlikely.

### 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

Ex. 26.1: CIs give intervals for unknown parameters. (The CI is a $$68$$% anyway, not $$95$$%.)

Ex. 26.3. CIs gives interval for the population mean, not for the data.

### 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. 27.9. 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. 31.5. $$H_0$$: $$p = 0.0602$$ 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. 35.5: 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. 36.5: 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

Ex. 37.1. Summary of mean differences; histogram of differences. Paired samples $$t$$-test; CI for mean difference.

Ex. 37.3. Comparing two odds: odds ratios; stacked or side-by-side bar chart. A $$\chi^2$$-test; CI for odds ratio.

### 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.

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.