E Answers to end-of-chapter exercises
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. 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. 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).
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. 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. 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. Probably in case hive size is a confounder.
Ex. 7.7. Statements 1, 3, 4, 8 and 9 true. 'Sex', 'Initial weight' possible confounders.
Ex. 7.9. 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. 7.11.
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. 7.13. Carry-over effect; observer effect.
Ex. 7.15. No; also possible in observational studies.
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. 8.10. 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. 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. 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. No answer. 3. No answer. 4. No answer. 5. 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. 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. 12.5. Bar (or dot) chart. Pie chart inappropriate: more than one option can be selected.
Ex. 12.7. Plots not shown. 4. Advantage: availability. Disadvantage: high price. 5. Table not shown. 6. \(0.209\). 7. \(1.21\).
Ex. 12.9. \(39.1%\). \(28.7\)%. \(0.642\). \(0.402\). \(1.6\). Table not shown.
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. \(0.354\).
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.
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.'
Ex. 26.11. CI: \(-49.36\) to \(-27.88\) mg.dl\(^{-1}\). Meaning, interpretation same as in Sect. 26.7.
<-- ::: {.answer data-latex=""} Ex. 26.12. CI: \(0.63\) kg to \(1.09\) kg (approx); \(0.63\) to \(1.09\) kg. Very similar. Stat. valid. \(-1.09\) to \(-0.63\) kg. Probably not.
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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.