E Answers to end-of-chapter exercises
Chap. 1: Introduction
Chap. 2: RQs
Ex. 2.1. 1. Percentage of vehicles that crash. 2. Average jump height. 3. Average number of tomatoes per plant.
Ex. 2.3. 1. Type of diet. 2. Whether coffee is caffeinated or decaffeinated. 3. Number iron tablets per day.
Ex. 2.5. 1. Between-individuals. Outcome: percentage wearing hats. 2. Between-intervals. Outcome: average yield (in kg/plant, tomatoes/plant, etc).
Ex. 2.7. 1. Correlational. 2. No sense assigning one variable as explanatory, another response.
Ex. 2.9. 1. P: Danish University students; O: Average resting diastolic blood pressure; C: between students who regularly drive, ride their bicycles to uni. 2. No intervention. 3. Relational. 4. Decision-making. 5. Conceptual: 'regularly'; 'university student' (on-campus? undergraduate? full-time?). Operational: how 'resting diastolic blood pressure' measured. 6. Resting diastolic blood pressure; whether they regularly drive, ride to uni. 7. Danish university students; Danish university students.
Ex. 2.11. 1. Probably relational. 2. Two-tailed. 3. Probably not. 4. How individual people using phones ('Talking'; 'texting'). 5. Walking speed. 6. Average walking speed.
Ex. 2.13. 1. Animal. 2. Pen: food is allocated to pen. Animals in the same pen are not independent: they compete for the same space, food, resources, and have similar environments. 3. Between diets.
Ex. 2.15. The \(10\) adults is sample. Unclear how many fonts compared (or which fonts). Perhaps: 'Among Australian adults, is the average time taken to read a passage of text different when Arial font is used compared to Times Roman font?'
Ex. 2.17. 1. Analysis: person; observation: individual nose hairs. Each unit of analysis has \(50\) units of observation. 2. \(n = 2\).
Ex. 2.19. 1. P: American adults; individuals: American adults. 2. O: average number of recorded steps. 3. Response: number of steps recorded for individuals. Explanatory: location of accelerometer. 4. Within individuals.
Ex. 2.21. 1. Relational; decision-making. 2. Correlational; estimation.
Unclear if intervention; seems unlikely.
Ex. 2.23. \(n = 27\); emu is unit of analysis.
Ex. 2.25. 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. 2.27. The board. Five units of analysis. Ten. Ten. Within-board variation much smaller (apart from first board).
Chap. 3: Overview of research design
Ex. 3.1. 1. Arsenic concentration. 2. Distance of lake from mine. 3. No: recorded; cannot be lurking. 4. Yes: may be related to the response, explanatory variables. 5. Confounding variable. 6. Observational: researchers do not determine the distance of lakes from mine.
Ex. 3.3. Response: perhaps 'risk of developing a cancer of the digestive system'. Explanatory: 'whether or not the participants drank green tea at least three times a week'. Lurking: 'health consciousness of the participants' (appears unrecorded).
Ex. 3.5. True; false; false; false; false; true. False. True.
Ex. 3.7. Age of the person.
Ex. 3.9. 1. Response variable. 2. Possible confounder. 3. Certain climate is needed to grow wheat; possible confounder? 4. Irrelevant. 5. Farm size: likely confounder (larger farms more efficient, may produce better yield). 6. Hours of sunlight: extraneous but not confounding perhaps?
Control: perhaps farm size (e.g., only farms over a certain size).
Ex. 4.3. True experiment.
Ex. 4.5. Quasi-experiment.
Ex. 4.7. 1. Many answers possible. 2. Researchers intervene: researchers give or not give subjects a pet. 3. Researchers do not intervene: find the subjects who do or do not already own a pet.
Chap. 6: Sampling
Ex. 6.1. c. Externally-valid study more likely.
Ex. 6.3. 1. Every \(7\)th day is same day of week. 2. Maybe select days at random over three-months.
Ex. 6.5. 1. Multi-stage. 2. Stratified (selecting floor), then convenience. 3. Convenience. 4. Part stratified (selecting floors), then convenience. First might be best.
Ex. 6.7. Random sampling to select schools. Then, self-selecting.
Ex. 6.9. Stratified: zones are strata.
Ex. 6.11. In Stage \(2\), selection of farms not random.
Ex. 6.13. 1. Households in Santiago. 2. ...if the sample is representative of all households in Santiago. 3. Voluntary response. 4.Multi-stage.
Chap. 7: Internal validity
Ex. 7.1. All are false.
Ex. 7.3. Yes; yes; yes; yes; yes; no (external validity).
Ex. 7.5. Also possible in observational studies.
Ex. 7.7. Probably in case hive size is a confounder.
Ex. 7.9. Statements 1, 3, 4, 8 and 9 true. 'Sex', 'Initial weight' possible confounders.
Ex. 7.11. 1. Observational. 2. Response: amount of sunscreen used; explanatory: time applying sunscreen. 3. Potential confounding variables. 4. If the mean of both the response and explanatory variables was different for females and males, sex of the participant a confounder; would need to be factored into the data analysis. 5. Participants blinded to what is happening in study.
Ex. 7.13.
1. Randomly allocate type of water to subjects (or the order subjects taste each drink.) 2. Subjects do not know which type of water they are drinking. 3. Person providing water and receiving ratings does not know which type of water subjects drinking. 4. Hard to find a control. 5. Any random sampling is good, if possible.
Observer effect: researcher directly contacting the subjects; may unintentionally influence responses.
Ex. 7.15. Carry-over effect; observer effect.
Chap. 8: Research design limitations
Ex. 8.1. External.
Ex. 8.3. Population: 'on-campus university students where (I) work'. External validity: whether the results apply to other members of target population.
Ex. 8.5. Sample not random; the researchers (rightly) state that results may not generalise to all hospitals. Because data only collected at night, perhaps not ecologically valid.
Ex. 8.7. Observational study: people with severe cough may take more cough drops.
Ex. 8.9. Study lacks ecological validity.
Chap. 9: Collecting data
Ex. 9.1. No place for \(18\)-year-olds.
Ex. 9.3. Best: second. First: leading (concerned cat owners...) Third: leading (Do you agree...)
Ex. 9.5. First fine; 'seldom' (for instance) may mean different things to different people; possible recall bias. Second: overlapping options (both \(1\)and \(2\)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. Sex of the person
Ex. 10.9. 1. Blood pressure: quant. continuous. 2. Program: qual. nominal. 3. Grade: qual. ordinal. 4. Number of doctor visits: quant. discrete.
Ex. 10.11. 1. Qual. nominal. 2. Quant. discrete. 3. Qual. ordinal (perhaps quant. discrete). 4. Qual. nominal. 5. Quant. continuous.
Ex. 10.13. Gender: qual. nominal. Age: quant. continuous. Height: quant. continuous. Weight: quant. continuous. GMFCS: qual. ordinal.
Ex. 10.15. 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 \(1.5\)/L. Most between \(4\) and \(3\)/L. Slightly right skewed. Some large outliers.
Probably the median as slightly skewed right, but with some outliers. Both the mean and median can be quoted.
Ex. 11.5. 1. \(3.7\). 2. \(3.5\). 3. \(1.888562\).
Ex. 11.7. Stemplot not shown. 1. Mean: \(-2.42\). 2. Median: \(0.8\). 3. Range: \(29.6\) (from \(-19.8\) to \(9.8\)). 4. Std. dev.: \(9.831172\); about \(9.83\). 5. IQR: \(4.95 - (-11.4) = 16.35\) (not including the median in each half). (No units of measurement.)
Ex. 11.9. 1. In order (in cm): \(127.4\); \(129.0\); \(14.4\); \(24\) using my software. Manually (without median in each half): \(Q_1 = 113\) and \(Q_3 = 138\) so IQR is \(25\). 2. We don't know. 3. No answer. 4. No answer. 5. No answer. 6. Hard to describe with standard language.
Ex. 11.11. No answer (yet).
Ex. 11.15. D; C; A; D.
Chap. 12: Summarising qualitative data
Ex. 12.1. Most common social group: many females plus offspring. No commonly-observed social group include males. Graph not shown.
Ex. 12.3. None are bad. I'd prefer bar chart, but any OK.
Ex. 12.5. 1. Nominal: gender; ordinal: place of residence; responses. 2. Gender: modes are female and male. Place: City \(> 100\ 000\) residents. Response: Agree. 3. Gender: NA.Place: City \(20\ 000\) to \(100\ 000\) residents. Response: Neutral. 4. \(5.12\): respondents about five times more likely to come from city than rural. 5. \(0.613\): respondents about \(0.61\) times as likely to agree or strongly disagree than choose other option. 6. \(1\): respondents just as likely to be male as female.
Ex. 12.7. 1. Walking; Bus 2. Bus. 3. No. 4. \(3.44\); that is, students \(3.44\) times as likely to use motorised transport than active transport. 5. \(0.141\); that is, for every \(100\) students that do not walk to campus, about \(100\times 0.141 = 14.1\) do walk to campus. 6. Figure not shown. The left panel shows the specific methods, and the right panel shows the methods of transport grouped more coarsely.
Ex. 12.9. Age, FEV and Height: histogram. Gender: bar or dot; mode: male (\(51.4\)%; odds: \(1.06\)). Smoking: bar or pie; mode: non-smoking (\(9.9\)%; odds: \(0.11\)).
Ex. 12.11. \(\text{OR(win; home)} = 4/6 = 0.6667\); \(\text{OR(win; away)} = 7/4 = 1.75\). \(\text{OR} = 0.6667/1.75 = 0.381\).
Chap. 13: Qualitative data: Comparing between individuals
Ex. 13.1. 1. Vomited: \(0.50\) beer then wine; \(0.50\) wine only. Didn't vomit: \(0.738\) beer then wine, \(0.262\) wine only. Prop. that drank various things, among those who did and didn't vomit. 2. Beer then wine: \(8.8\)% vomited, \(91.2\)% didn't; Wine only: \(21.4\)% vomited, \(78.6\)% didn't. Percentage that vomited, for each drinking type. 3. \((6 + 6)/(6 + 6 + 62 + 22) = 0.125\). 4. \(0.2727\). 5. \(0.096774\). 6. \(2.82\). 7. \(0.354\). 8. \(-0.176\). 9. Higher percentage vomited after beer-then-wine, compared to beer only.
Ex. 13.3. 1. About \(18.4\)%. 2. About \(25.9\)%. 3. About \(11.7\)%. 4. About \(0.226\). 5. \(0.35\). 6. About \(0.132\). 7. About \(2.7\). 8. Odds no August rainfall in Emerald \(2.7\) times higher in months with non-positive SOI.
Ex. 13.5. Plot not shown.
Ex. 13.7. 1. Prop. F skipped: \(\hat{p}_F = 0.359\); 2. Prop. M skipped: \(\hat{p}_M = 0.284\). 3. Odds(Skips breakfast, F): \(0.5598\); 4. Odds(Skips breakfast, M): \(0.3966\). 5. Odds ratio: \(1.41\). 6. Odds females skipping are \(1.41\) times the odds males skipping. 7. Not shown.
Ex. 13.9. \(39.1%\). \(28.7\)%. \(0.642\). \(0.402\). \(1.6\). Table not shown.
Chap. 14: Quantitative data: Comparing within individuals
Chap. 15: Quantitative data: Comparing between individuals
Ex. 15.1. 1. In general, DB has smaller cost over-runs. 2. Hard to tell: DB: \(2\); DBB: \(3\). 3. Hard to tell: DB: \(2\); DBB: \(3\).
Ex. 15.3. A. II (median; IQR). B. I (mean; standard deviation). C. III (median; IQR).
Ex. 15.5. 1. \(0.61\); \(0.40\); \(0.42\) panels per min. 4. Worker 2 faster, more consistent (using IQR); Worker 1 slower. Plots not shown.
Ex. 15.7. No answer (yet).
Ex. 15.8. 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.
Chap. 16: Quantitative data: Correlations between individuals
Ex. 16.1. Many correct answers.
Ex. 16.3. You cannot be precise. From software: \(r = 0.71\). The best you can do is 'a reasonably high, positive \(r\) value'.
Ex. 16.5. 1. A tree. 2. Form: starts straight-ish, then hard to describe. Direction: biomass increases as age increases (on average). Variation: small-ish for small ages; large-ish for older trees (after about \(60\)).
Ex. 16.7. Approximately linear; positive relationship; variation seems to get larger for a larger number of cases.
Ex. 16.9. No answer (yet).
Ex. 16.11. No answer (yet).
Ex. 16.13. \(R^2 = (-0.682)^2 = 0.465\): about \(46.5\)% of the variation in the number of cyclones explained by knowing value of ONI averaged over Oct. to Dec.; extraneous variables explain the remaining \(54.5\)% of the variation in the number of cyclones.
Chap. 17: More about summarising data
Ex. 17.1. Scatterplot; histogram of the diffs; side-by-side bar.
Ex. 17.3. Individual variables: bar chart for origin; histogram for others. Between biomass, origin: boxplot. Between biomass, other variables: scatterplot. (On scatterplot, origins could be encoded with different colours or symbols.)
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. 1. Just Kings and Aces. 2. \(8/52 = 2/13\). 3. Picture cards. 4. \(16/52 = 4/13\). 5. Picture card, spade at same time. 6. \(4/52 = 1/13\). 7. Any \(\heartsuit\), \(\diamondsuit\) or \(\clubsuit\). 8. \(39/52 = 3/4\). 9. \(4/16 = 1/4\). 10. \(4/13\).
Ex. 18.5. False. True. \(1/2\). \(1/2\). HH, HT, TH, TT (Coin A listed first).
Ex. 18.7. \(4/6\). \(5\). Yes: what happens on die won't change coin outcome. \(1/2\). \(1/6\). \(1/3\).
Ex. 18.9. 1. In order drawn: BB, BR, RB, RR. 2. Equally-likely outcomes: \(1/2\). 3. \(1/2\). 4. Yes.
Ex. 18.11. No answer (yet).
Ex. 18.13. 1. Not independent: If it rains, less likely to walk to work than if it doesn't rain. 2. Not independent: A smoker is far more likely to suffer from lung cancer than a non-smoker. 3. Dependent: If it rains, I won't water my garden.
Ex. 18.15. The reasoning assumes the three outcomes (HH, TT, HT) equally likely, which is not true. For example, consider tossing a \(20\)-cent coin (shown in lower-case, normal font) and a \(1\)-dollar coin (shown in capitals, bold font). The four outcomes are: hH, hT, tH tT.
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. \(68.26\)%; very close to \(68\)%.
Ex. 21.9. 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. 21.11. About \(z = 2.05\). IQ: \(130.75\). An IQ greater than about \(130\) is required to join Mensa.
Ex. 21.13. Use number of minutes from (say) 5:30pm. Std. dev.: \(120\), plus \(0.28\times 60 = 16.8\)= \(136.8\). 1. \(9\)pm; \(210\)from \(5\):\(30\)pm; \(z = 1.54\); about \(6.2\)%. 2. \(z = -0.22\); \(41.3\)%. 3. \(z_1 = -0.22\) and \(z_2 = 0.22\); \(0.5871 - 0.4129\); about \(17.4\)%. 4. \(z\)-score: \(0.52\); \(x = 71.136\) minutes after \(5\)pm; about one hour and \(11\) mins after \(5\):\(30\)pm, or \(6\):\(41\)pm. 5. \(z\)-score: \(-1.04\); \(x = -141.272\), or \(141.272\) mins before \(5\):\(30\)pm; about two hours and \(21\)before \(5\):\(30\)pm, or \(3\):\(09\)pm.
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. \(\hat{p} = 0.241\). \(\text{s.e.}(\hat{p}) = 0.01098449\). 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. Normal; mean \(\mu\); std dev: \(3.757\), 4. \(77.4\) to \(92.4\). 5. Approx. \(95\)% confident the population mean weight of male American black bears between \(77.4\) and \(92.4\). 6. Stat. valid: \(n \ge 25\).
Ex. 24.3. \(\text{s.e.} = 0.06410062\). Approx. \(95\)% CI: \(2.72\)to \(2.98\).
Ex. 24.5. Approx. \(95\)% CI : \(29.9\) s to \(36.1\) s.
Ex. 24.7. None acceptable. 1. CIs not about observations, but statistics. 2. CIs not about observations, but statistics. 3. Samples don't vary between values; statistics do. (CIs about populations anyway.) 4. Populations don't vary between values. 5. Parameters do not vary. 6. We know \(\bar{x} = 1.3649\)/L. 7. We know \(\bar{x} = 1.3649\)/L.
Ex. 24.9. \(\text{s.e.}(\bar{x}) = 5.36768\); approx. \(95\)% CI: \(50.56\) s to \(72.04\) s. Stat. valid.
Ex. 24.11. 1. One observation is \(x = 44\); the claimed population mean is \(\mu = 45\). 2. Fine to have decimal value as a mean. 3. \(\bar{x} = 44.9\); \(\mu = 45\): two different things; why should they be the same? 4. What two broad reasons could explain why the sample mean is not \(45\)? 5. CI allows for sampling variation. 6. \(44.8504\) to \(44.9496\). 7. Possibly lying; not certain. 8. \(x = 44\), \(\bar{x} = 44.9\), \(\mu = 45\), \(s = 0.124\).
Chap. 25: More about CIs
Ex. 25.1. 1. CIs give intervals for unknown parameters, not known statistics. 2. CIs for the proportion (or percentage), not number of trees. (The CI is \(68\)% anyway, not \(95\)%.)
Ex. 25.3. 1. CIs not about individuals. 2. CIs not about sample means.
Ex. 25.5. Intervals for different things. First: \(95\)%CI for mean weight. Second: not a CI; for weights of individuals possums.
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 \(2.230\)(\(\text{s.e.} = 10.879\); \(n = 23\)), with an approx. \(95\)% CI between \(24.79\)lighter in 2014 to \(20.33\)higher in 2015. 5. Probably; \(n\) just less than \(25\). 6. Pairs have same farm management, soil, etc.
Ex. 26.5. 1. Table not shown. 2. Approx. \(95\)% CI: \(-0.92\) to \(11.22\). 3. How much tastier Broccoli is with dip.
Ex. 26.7. \(1.65\) to \(4.37\) pounds. Possibly not practically important.
Ex. 26.11. 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 \(9.71\). 'In the population, the mean difference between the amount of vigorous PA by Spanish health students is between \(4.35\)more during lockdown, and \(9.71\)more before lockdown.'
Chap. 27: CIs for two means
Ex. 27.1. 1. Parameter: \(\mu_M - \mu_F\). Estimate: \(\bar{x}_M - \bar{x}_F = -0.06\). 2. Plot not shown. 3. \(0.0928735\) 4. Table not shown. 5. \(-0.25\) to \(0.13\). 'The population difference between the mean lengths of female and male gray whales at birth has a \(95\)% chance of being between \(0.13\), longer for male whales to \(0.25\)longer for female whales.' 6. 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. 8. Probably not practically important (diff. between the means: average of \(0.53\) days).
Ex. 27.5. Normal; mean at \(\mu_N - \mu_P\); std. dev.: \(19.612\).
Ex. 27.6. Normal; mean at \(\mu_B - \mu_A\); std. dev.: \(2.96\).
Ex. 27.7. \(11.9\) to \(22.5\), greater for industrialised countries.
Ex. 27.9. 1. Either direction fine; here \(\mu_Y - \mu_O\): the mean amount younger women can lean further forward than older. 2. Dot plot? 3. Table not shown. 4. Approx. CI: \(10.166\) to \(18.834\)oC. Exact CI (Row 2): \(9.10\) to \(19.90\)oC. 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. 27.11. No answer (yet).
Ex. 27.13. 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. \(50\times 52/172 = 15.1\); yes.
Ex. 28.5. Normal; mean \(p_P - p_N\) and std. dev. \(0.0428\). For OR: not normal distribution.
Ex. 28.7. 1. Individual std. errors: \(0.04191\); \(0.04019\); diff: \(0.05806\) 2. \(-0.2927\) to \(-0.0604\), or \(0.0604\) to \(0.2927\) greater for beach swimmers. 3. \(0.0627\) to \(0.2903\) greater for beach swimmers. 4. (No answer.) 5. Odds no infection: \(61/79 = 0.772\) (non-beach) and \(90/57 = 1.579\) (beach); OR: \(0.490\) as in output. 6. \(0.3054\) to \(0.7831\) (Non-beach vs beach). 7. (No answer.) 8. \(140\times136/287 = 66.3\); yes. 9. Table not shown.
Ex. 28.9. 1. Table not shown. 2. The individual standard errors: \(0.03446\) and \(0.06331\); for difference: \(0.07209\) 3. \(-0.2626\) to \(0.0258\), larger when run for \(3000\). 4. \(-0.2597\) to \(0.0229\), larger when run for \(3000\). 5. (No answer.) 6. \(0.1331\) to \(1.1366\). 7. (Not answer.) 8. \(42\times 16/115 = 5.84\); yes.
Ex. 28.11. 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. \(199\times 139/668 = 41.1\); yes.
Chap. 29: Estimating sample sizes
Ex. 29.1. 1. At least \(25\). 2. At least \(100\) (\(4\) times as many). 3. At least \(400\) (\(16\) times as many). 4. To halve width, need \(4\) times as many. 5. To quarter width, need \(16\) times as many.
Ex. 29.3. 1. At least \(10\ 000\). 2. At least \(2\ 500\). 3. At least \(1\ 000\) needed. 4. Expensive (time and money): \(10\ 000\) and \(2\ 500\) probably unrealistic.
Ex. 29.5. Use \(s = 0.43\). 1. At least \(1\ 849\). 2. At least \(2\ 96\). 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. In testing, we assume the value of \(p\) as stated in \(H_0\). In CIs, we have no value of \(p\).
Ex. 30.3. Tests are not about the sample value (we know the value of \(\hat{p}\)) in our sample), but about unknown population value (i.e., \(p\)).
Ex. 30.5. 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. 30.7. \(H_0\): \(p = 0.5\); \(H_1\): \(p \ne 0.5\). \(\hat{p} = 0.39726\); \(\text{s.e.}(\hat{p}) = 0.05727\); \(z = -1.794\). \(P\)-value not that small. No evidence of difference.
Ex. 30.9. \(H_0\): \(p = 0.0602\); \(H_1\): \(p < 0.602\) (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 that proportion of females using the machines was lower than the proportion of females in the university population.
Ex. 30.11. \(H_0\): \(p = 0.5\); \(H_1\): \(p > 0.5\) (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 majority like breadfruit pasta (for population represented by sample anyway).
Ex. 30.13. \(H_0\): \(p = 1/16 = 0.625\); \(H_1\): \(p \ne 0.0625\). \(\hat{p} = 0.1395349\) and \(\text{s.e.}(\hat{p}) = 0.01845701\); \(z = 26.3\): massive; the \(P\)-value incredibly small. Very strong evidence pop. proportion not \(1/16\); borers not resistant.
Ex. 30.15. \(\hat{p} = 0.56\); \(z = 1.91\). No evidence of a bias.
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 \(90\).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 \(14\).
SD: The variation in the weight of individual bars. SE: The variation in the sample means for samples of size \(67\).
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 \(\mu\). 2. RQ two-tailed. 3. \(36.8052\) is a sample mean; hypothesis can be written before data collected. 4. Hypotheses about parameters; \(36.8052\) is a sample mean. These hypotheses are asking to test if the sample mean is \(36.8052\)... which we know it is. 5. Hypothesis can be written down before data collected. 6. Hypotheses are about parameters.
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. 4. The \(P\)-value is \(0.018\), not less than \(0.018\).
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. Graph not shown. 2. Mean increase in fruit weight from normal to dry (i.e., normal minus dry). 3. \(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d \ne 0\). 4. Sketch the sampling distribution. 5. \(t = -0.205\). 6. \(P\) large; from software, \(P = 0.839\). 7. 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: \(2.230\)(\(95\)% CI from \(-24.8\) to \(20.3\)), heavier in normal year). **8.* \(n = 24\); probably.
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 females minus mean length males. 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: \(12.70\); standard deviation: \(0.611\)) and males (mean: \(12.07\); standard deviation: \(0.705\); \(95\)% CI for the difference: \(-1.26\)to \(0.246\)). 5. Stat. 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 \(22.54\)/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: \(41.8\)/person/year) and non-industrialised (mean: \(24.6\)/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: \(30.7\)oC; std. dev.: \(2.58\)oC; \(n = 10\)) compared to older women (mean: \(16.20\)oC; std. dev.: \(4.44\)oC; \(n = 5\); \(95\)% CI for the difference: \(9.1\)oC to \(19.9\)oC).
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: \(36.886\)oC) and males (mean: \(36.725\)oC). The diff. between the means, of \(0.16\) of a degree, of little practical importance.
Ex. 27.14. 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. 1. \(z = 3.27\). 2. Very small.
Ex. 35.5. 1. Table not shown. 2. Graph no shown. 3. Moderate evidence that the population odds of finding a male sandfly in eastern Panama is different at \(3\) ft above ground compared to \(35\) ft above ground. 4. Stat. valid.
Ex. 35.7. No answer (yet).
Ex. 35.9. 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. 35.11. \(\chi^2 = 22.374\), 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. 35.13. 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. 35.15. 1. to 3. Table not shown. 4. \(H_0\): No association; \(H_1\): Association exists. 5. \(\chi^2 = 16.24\); \(P < 0.001\). 6. Very strong evidence in the sample that proportion bringing a bag is not the same for all three age groups. 7. Smallest exp. count: \(24.7\); 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. 1. Larger width associated with larger heights: small-to-moderate correlation. 2. Correlation possibly zero. 3. Correlation possibly zero. 4. Need separate reports for men's, women's: no correlations.
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 \(1\)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 \(1\)increase in chest circumference. 5. Intercept: cm; slope: cm/gram. 6. \(\hat{y} = 2\ 538.7\).
Ex. 38.15. Plots not shown.
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. 39.7. Number decimal places ridiculous.
Ex. 39.9. 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. 39.11. 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.