Answers to odd-numbered exercises
Chap. 1: Research an introduction
Chap. 2: Research questions
Ex. 2.1. 1. Percentage of vehicles that crash. 2. Average jump height. 3. Average number of tomatoes per plant.
Ex. 2.3. 1. Diet type. 2. Whether coffee is caffeinated, decaffeinated. 3. Num. iron tablets/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 variables as explanatory, response.
Ex. 2.9. 1. P: Danish Uni 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?). Operational: how 'resting diastolic blood pressure' measured. 6. Resting diastolic blood pressure; whether they regularly drive, ride to uni. 7. Both: Danish uni 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. 7. Observation: person. Analysis: individual people not in a group, since people in a group may not walk independently of each other (i.e., they keep pace with each other).
Ex. 2.13. 1. Animal. 2. Pen: food allocated to pen. Animals in same pen not independent: compete for same space, food, resources, have similar environments. 3. Between diets.
Ex. 2.15. Ten adults is sample. Unclear how many (or which) fonts compared. Perhaps: 'Among Australian adults, is average time taken to read passage of text diff 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 recorded steps. 3. Response: number steps recorded for individuals. Explanatory: location of accelerometer. 4. Within individuals.
Ex. 2.21. 1. Relational; decision-making. 2. Correlational; estimation. Intervention unlikely.
Ex. 2.23. \(n = 27\); unit of analysis: emu.
Ex. 2.25. Unit of observation: tyre. Unit of analysis: car. Brand allocated to car; each car gets one tyre brand. Tyres on any car exposed to same day-to-day use, drivers, distances, etc. Each unit of analysis produces four units of observations. Sample size: \(10\) cars (\(40\) observations).
Ex. 2.27. Board. Five units of analysis. Ten. Ten. Within-board variation smaller (except 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 response, explanatory variables. 5. Confounding variable.
Ex. 3.3. Response: perhaps 'risk of developing cancer of digestive system'. Explanatory: 'whether participants drank green tea at least \(3\) times per week'. Lurking: 'health consciousness of participants' (appears unrecorded).
Ex. 3.5. T; F; F; F; F; T. F. T.
Ex. 3.7. Age of person.
Ex. 3.9. 1. Response. 2. Possible confounder. 3. Certain climate needed to grow wheat; possible confounder? 4. Irrelevant. 5. Farm size: likely confounder (larger farms more efficient, may produce better yield). 6. Hours sunlight: extraneous but perhaps not confounding. Control: perhaps farm size (e.g., farms over certain size).
Chap. 4: Types of research studies
Ex. 4.1. 1. Between-individuals. 2. Relational. 3. Most likely. 4. Estimation. 5. Intervention: experiment. Likely true experiment.
Ex. 4.3. True experiment.
Ex. 4.5. Quasi-experiment.
Ex. 4.7. 1. Answers vary. 2. Researchers intervene: researchers give, not give subjects pet. 3. Researchers do not intervene: find the subjects who do, do not already own pet.
Chap. 6: External validity: sampling
Ex. 6.1. c. Externally-valid study more likely.
Ex. 6.3. 1. Under: practicality; unintentional. 2. Under; deception; probably intentional. 3. Over; deception; intentional (cherry-picking).
Ex. 6.5. 1. Every \(7\)th day is same day of week. 2. Maybe select days at random over three-months.
Ex. 6.7. 1. Multi-stage. 2. Stratified (floor), then convenience. 3. Convenience. 4. Part stratified (floors), then convenience. First perhaps?
Ex. 6.9. Random sampling to select schools. Then, self-selecting.
Ex. 6.11. Stratified: zones are strata.
Ex. 6.13. Stage \(2\) selection of farms not random.
Ex. 6.15. 1. Households in Santiago. 2. ...if sample representative of all households in Santiago. 3. Voluntary response. 4. Multi-stage.
Chap. 7: Internal validity
Ex. 7.1. All false.
Ex. 7.3. Yes; yes; yes; yes; yes; no (external).
Ex. 7.5. Also possible in observational studies.
Ex. 7.7. In case hive size 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 sunscreen used; explanatory: time applying sunscreen. 3. Potential confounding variables. 4. If mean of both response, explanatory variables different for F and M, sex a confounder; would need to be factored into the analysis. 5. Participants blinded.
Ex. 7.13. Random allocation; exclusion criteria; blinding; comparing two groups; ethical.
Ex. 7.15. 1. A: no blinding; B: double-blind. 2. Hawthorne effect impacting internal validity. 3. B; no Hawthorne effect.
Ex. 7.16. 1. Subjective: accelerometer. 2. Perceptions unreliable; Hawthorne effect. 3. Using accelerometer.
Ex. 7.17. 1. Randomly allocate type of water to subjects (or order subjects taste each drink.) 2. Subjects do not know which type of water they drink. 3. Person providing water and receiving ratings does not know which type of water subjects drink. 4. Hard to find control. 5. Any random sampling is good, if possible. Observer effect: researcher directly contacting the subjects; may unintentionally influence responses.
Ex. 7.19. 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 results apply to other members of target population.
Ex. 8.5. Sample not random; researchers (rightly) state results may not generalise to all hospitals. Data collected at night; not ecologically valid?
Ex. 8.7. Observational study: people with severe cough may take more cough drops.
Ex. 8.9. 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 \(1\,\text{h}\) and \(2\,\text{h}\) in two categories).
Chap. 10: Classifying data and variables
Ex. 10.1. Quant. continuous. Qual. nominal. Quant. continuous. Qual. nominal.
Ex. 10.3. F, T, F.
Ex. 10.5. Nominal; qualitative.
Ex. 10.7. Sex of person
Ex. 10.9. 1. Quant. continuous. 2. Qual. nominal. 3. Qual. ordinal. 4. 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.; four values used; probably treated as qual. ordinal. Mob size: quant. discrete. Sex: qual. nominal.
Chap. 11: Summarising quantitative data
Ex. 11.1. Shape: skewed left. Average: perhaps \(70\)--\(80\)? Variation: most between \(30\), \(80\). Outliers: none; 'bump' at lower ages.
Ex. 11.3. Slightly right skewed. Average near \(1.5\,\text{mmol}\)/L. Most between \(3\), \(4\,\text{mmol}\)/L. Some large outliers. Probably median (slightly skewed right; outliers). Both mean and median can be quoted.
Ex. 11.5. 1. \(3.7\). 2. \(3.5\). 3. \(1.888562\). 4. \(5 - 2 = 3\).
Ex. 11.7. Plot not shown. 1. Mean: \(-2.42\). 2. Median: \(0.8\). 3. Range: \(29.6\) (\(-19.8\) to \(9.8\)). 4. Std dev.: \(9.831172\); about \(9.83\). 5. IQR: \(4.95 - (-11.4) = 16.35\) (not including median in each half). (No units of measurement.)
Ex. 11.9. 1. In cm: \(127.4\); \(129.0\); \(14.4\); \(24\) from software. Manually (without median in each half): \(Q_1 = 113\), \(Q_3 = 138\), IQR is \(25\). 2. Don't know. 3.--5. Not shown. 6. Hard to describe with standard language.
Ex. 11.11. Average: about \(10\)?; variation: \(0\) to \(30\) perhaps; skewed right. Value between \(35\) and \(40\) perhaps outlier.
Ex. 11.13. 1. Men's: about \(50\)%; women's: about \(100\)%. 2. Men's: about \(0\)%; women's: about \(50\)%.
Ex. 11.15. D; C; A; D.
Chap. 12: Summarising qualitative data
Ex. 12.1. Most common social group: many F plus offspring. No commonly-observed social group include M. Graph not shown.
Ex. 12.3. None bad. I'd prefer bar chart; any OK.
Ex. 12.5. 1. Nominal: gender; ordinal: place of residence; responses. 2. Gender: modes are F, M. Place: City \(> 100\,000\) residents. Response: Agree. 3. Gender: NA.Place: City \(20\,000\) to \(100\,000\) residents. Response: Neutral. 4. \(5.12\): respondents about \(5\) times more likely to come from city than rural. 5. \(0.613\): respondents about \(0.61\) times as likely to agree, strongly disagree than choose other option. 6. \(1\): respondents as likely to be M as F.
Ex. 12.7. 1. Walking; Bus 2. Bus. 3. No. 4. \(3.44\); i.e., students \(3.44\) times as likely to use motorised transport than active. 5. \(0.141\); i.e, for every \(100\) students that do not walk, \(100\times 0.141 = 14.1\) do walk. 6. Not shown. Left panel shows specific methods, right methods of transport grouped more coarsely.
Ex. 12.9. Age, FEV, Height: histogram. Gender: bar or dot; mode: M (\(51.4\)%; 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: Comparing quantitative data within individuals
Chap. 14: Comparing quantitative data between individuals
Ex. 14.1. 1. In general, DB has smaller cost over-runs. 2. Hard to tell: DB: \(2\,\text{cm}\); DBB: \(3\,\text{cm}\). 3. Hard to tell: DB: \(2\,\text{cm}\); DBB: \(3\,\text{cm}\).
Ex. 14.3. A. II (median; IQR). B. I (mean; standard deviation). C. III (median; IQR).
Ex. 14.5. 1. \(0.61\); \(0.40\); \(0.42\) panels/min. 4. Worker 2 faster, more consistent (using IQR); Worker 1 slower. Plots not shown.
Ex. 14.7. 1. Error bar chart. 2. Not shown.
Ex. 14.9. 1. Not shown. 2. Not shown.
Ex. 14.11.
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. Not shown.
3. Little diff between M, F in sample.
Ex. 14.13. 1. Very similar mean SVL. 2. Crayfish regions: smaller mean SVL. 3. Crayfish regions: larger mean SVL. 4. Confounding.
Chap. 15: Comparing qualitative data between individuals
Ex. 15.1. One.
Ex. 15.3. 1. Vomited: \(0.50\) beer, wine; \(0.50\) wine only. Didn't vomit: \(0.738\) beer, wine, \(0.262\) wine only. Prop. drank various things, among those who did, didn't vomit. 2. Beer, 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, wine, compared to beer only.
Ex. 15.5. 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. 15.7. Not shown.
Ex. 15.9. 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 F skipping \(1.41\) times odds M skipping. 7. Not shown.
Ex. 15.11. 1. Not shown. 2. \(74.6%\). 3. \(60.9\)%. 4. \(2.487\). 5. \(1.558\). 6. \(1.596\). 7. \(0.626\). 8. Not shown.
Chap. 16: Correlations between quantitative variables
Ex. 16.1. Answers vary.
Ex. 16.3. You cannot be precise. Software: \(r = 0.71\). Realistically: 'reasonably high, positive \(r\)'.
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 \(60\)).
Ex. 16.7. Approximately linear; positive relationship; variation larger for more cases.
Ex. 16.9. No relationship.
Ex. 16.11. Approx. linear; positive; strong.
Ex. 16.13. \(R^2 = (-0.682)^2 = 0.465\): about \(46.5\)% of the variation in number cyclones explained by knowing value of ONI; extraneous variables explain remaining \(54.5\)% of variation in number cyclones.
Chap. 17: More details about tables and graphs data
Ex. 17.1. Scatterplot; histogram of 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, could encode origins with different colours, 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. cont.). 2. Explanatory: treatment group (qual. nominal, \(3\) levels). 3. Response: histogram. Explanatory: bar chart. Relationship: boxplot.
Ex. 17.9. Plots not shown. Speed: average: around \(60\) wpm; variation: about \(30\) to \(120\) wpm. Slightly right skewed; no obvious outliers. Accuracy: average: around \(85\)%; variation: about \(65\)% to \(95\)%. Left skewed; no obvious outliers. Age: average: \(25\); variation: about \(15\) to \(35\). Very right skewed, perhaps unseen large outliers. Sex: about twice as many F as M. Speed and Sex: not big difference between M, F. Accuracy and Age: hard to see relationship; no older people 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. 1. \(0.087\). 2. \(0.708\). 3. F: prob FN: \(0.107\); M: prob FN: \(0.108\); close to independent. 4. F: prob FN: \(0.040\); M: prob FN: \(0.035\); close to independent. 5. Gov: prob FN: \(0.107\); NGov: prob FN: \(0.040\); not independent. 6. Gov: prob FN: \(0.108\); NGov: prob FN: \(0.035\); not independent. 7. Regardless of sex, First Nations children more likely to be at government school.
Ex. 18.13. 1. Not independent: Less likely to walk in rain. 2. Not independent: Smoker far more likely to suffer from lung cancer than non-smoker. 3. Dependent: If it rains, I won't water garden.
Ex. 18.15. Reasoning assumes three equally likely outcomes (HH, TT, HT); untrue. Consider tossing \(20\)-c coin (lower-case, normal) and \(1\)-coin (capitals, bold). Four outcomes: hH, hT, tH tT.
Chap. 19: Sampling variation
Chap. 20: Models and normal distributions
Ex. 20.1. All false.
Ex. 20.3. 1: C; 2: A; 3: B; 4: D.
Ex. 20.5. \(68.26\)%; very close to \(68\)%.
Ex. 20.7. 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. About \(z = 1.04\); tree diameter about \(11.6\) inches.
Ex. 20.9. 1. \(z = -0.61\); \(72.9\)%. 2. \(z = -1.83\); \(3.4\)%. 3. \(z = -4.878\) and \(z = -1.83\); \(3.4\)%. 4. \(z = 1.64\) (or \(1.65\)). \(5\)% longer than \(42.7\) weeks. 5. \(z\)-score: \(-1.28\). \(10\)% shorter than \(37.9\) weeks.
Ex. 20.11. \(z = 2.05\). IQ: \(130.75\). \(\text{IQ} > 130\).
Ex. 20.13. Use number minutes from (say) 5:30pm. Std dev.: \(120\,\text{mins}\), plus \(0.28\times 60 = 16.8\,\text{mins}\) = \(136.8\,\text{mins}\). 1. \(9\)pm; \(210\)mins$ from \(5\):\(30\)pm; \(z = 1.54\); \(6.2\)%. 2. \(z = -0.22\); \(41.3\)%. 3. \(z_1 = -0.22\) and \(z_2 = 0.22\); \(0.5871 - 0.4129\); \(17.4\)%. 4. \(z\)-score: \(0.52\); \(x = 71.136\) mins 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 \(141.272\,\text{mins}\) before \(5\):\(30\)pm; about two hours and \(21\)before \(5\):\(30\)pm, or \(3\):\(09\)pm.
Chap. 22: Confidence intervals: one proportion
Ex. 22.1. \(\hat{p} = 0.81944\), \(n = 864\). \(\text{s.e.}(\hat{p}) = 0.01309\); approx. \(95\)% CI: \(0.819 \pm (2\times 0.0131)\). Stat. valid.
Ex. 22.3. \(\hat{p} = 0.051948\); \(\text{s.e.}(\hat{p}) = 0.00178\); approx. \(95\)% CI: \(0.0519\pm 0.0358\). Stat. valid.
Ex. 22.5. \(\hat{p} = 0.317059\); \(n = 6882\). \(\text{s.e.}(\hat{p}) = 0.0056092\). CI: \(0.317\pm 0.011\). Stat. valid.
Ex. 22.7. \(\hat{p} = 0.241\). \(\text{s.e.}(\hat{p}) = 0.010984\). Approx. \(95\)% CI: \(0.219\) to \(0.263\).
Ex. 22.9. \(\hat{p} = 0.3182\). \(\text{s.e.}(\hat{p}) = 0.0702175\). Approx. \(95\)% CI: \(0.178\) to \(0.459\).
Ex. 22.11. \(\hat{p} = 0.13431\). \(\text{s.e.}(\hat{p}) = 0.012434\). Approx. \(95\)% CI: \(0.109\) to \(0.159\).
Chap. 23: Confidence intervals: one mean
Ex. 23.1. 1. Parameter: pop. mean weight of American black bear, \(\mu\). 2. \(\text{s.e.}(\bar{x}) = 3.756947\). 3. Normal; mean \(\mu\); std dev: \(3.757\), 4. \(77.4\) to \(92.4\,\text{kg}\). 5. Approx. \(95\)% confident population mean weight of male American black bears between \(77.4\) and \(92.4\,\text{kg}\). 6. Stat. valid: \(n \ge 25\).
Ex. 23.3. \(\text{s.e.} = 0.06410\). Approx. \(95\)% CI: \(2.72\,\text{L}\) to \(2.98\,\text{L}\).
Ex. 23.5. Approx. \(95\)% CI: \(29.9\) to \(36.1\,\text{s}\).
Ex. 23.7. 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. Know \(\bar{x} = 1.3649\,\text{mmol}\)/L. 7. Know \(\bar{x} = 1.3649\,\text{mmol}\)/L.
Ex. 23.9. \(\text{s.e.}(\bar{x}) = 5.36768\); approx. \(95\)% CI: \(50.56\) to \(72.04\,\text{s}\). Stat. valid.
Ex. 23.11. 1. One observation \(x = 44\); claimed population mean is \(\mu = 45\). 2. OK to have decimal value as mean. 3. \(\bar{x} = 44.9\); \(\mu = 45\): different things; why should they be same? 4. Sampling variation; real difference. 5. CI allows for sampling variation. 6. \(44.850\) to \(44.950\). 7. Possibly lying; not certain. 8. \(x = 44\), \(\bar{x} = 44.9\), \(\mu = 45\), \(s = 0.124\).
Chap. 24: More details about CIs
Ex. 24.1. 1. CIs give intervals for unknown parameters, not known statistics. 2. CIs for proportion (or percentage), not number of trees. (The CI is \(68\)% anyway, not \(95\)%.)
Ex. 24.3. 1. CIs not about individuals. 2. CIs not about sample means.
Ex. 24.5. Intervals for different things. First: \(95\)% CI for mean weight. Second: not CI; for weights of individuals possums.
Chap. 25: Making decisions
Chap. 26: Hypothesis tests: one proportion
Ex. 26.1. In tests, assume \(p\) known. In CI, have no value for \(p\) to use.
Ex. 26.3. \(0.38\) is sample proportion; RQ asks about pop. proportion of \(1/6\).
Ex. 26.5. Tests not about sample value (we know value of \(\hat{p}\)), but about unknown pop. value (i.e., \(p\)).
Ex. 26.7. 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. \(\hat{p} = 0.6173\); \(z = 9.39\): \(P\) very small: Very strong evidence people do better-than-guessing at identifying placebo.
Ex. 26.9. \(H_0\): \(p = 0.5\); \(H_1\): \(p \ne 0.5\). \(\hat{p} = 0.39726\); \(\text{s.e.}(\hat{p}) = 0.05727\); \(z = -1.794\). \(P\) not that small. No evidence of difference.
Ex. 26.11. \(H_0\): \(p = 0.0602\); \(H_1\): \(p < 0.602\) (one-tailed). \(\hat{p} = 0.5008489\); \(n = 589\): \(\text{s.e.}(\hat{p}) = 0.0201689\), so \(z = -5.015\). \(P\) very small. Strong evidence prop. F using machines lower than prop. F in uni pop.
Ex. 26.13. \(H_0\): \(p = 0.5\); \(H_1\): \(p > 0.5\) (one-tailed). \(\hat{p} = 0.802817\); \(n = 71\): \(\text{s.e.}(\hat{p}) = 0.0593391\), so \(z = 5.10\). \(P\) very small. Strong evidence majority like breadfruit pasta (for pop. represented by sample anyway).
Ex. 26.15. \(H_0\): \(p = 1/16 = 0.625\); \(H_1\): \(p \ne 0.0625\). \(\hat{p} = 0.139535\) and \(\text{s.e.}(\hat{p}) = 0.018457\); \(z = 26.3\): massive; \(P\) very small. Very strong evidence pop. proportion not \(1/16\); borers not resistant.
Ex. 26.17. \(\hat{p} = 0.56\); \(z = 1.91\). No evidence of bias.
Chap. 27: Hypothesis tests: one mean
Ex. 27.1. 1. \(\mu\), pop. mean speed (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) \(P\) very small. 7. Very strong evidence mean speed of vehicles on road greater than \(90\,\text{km}\).h\(-1\). 8. Stat. valid.
Ex. 27.3. \(H_0\): \(\mu = 50\); \(H_1\): \(\mu > 50\) (one-tailed). \(\text{s.e.}(\bar{x}) = 4.701076\). \(t = 7.23\): \(P\) very small. Very strong evidence (\(P < 0.001\)) mean mental demand greater than \(50\). (Greater than, because of RQ and alternative hypothesis.)
Ex. 27.5. \(H_0\): \(\mu = 14\); \(H_1\): \(\mu \ne 14\) (two-tailed). \(\text{s.e.}(\bar{x}) = 0.092493\). \(t\)-score: \(10.35\): huge; \(P\) very small. Very strong evidence (\(P < 0.001\)) mean weight of Fun Size Cherry Ripe bar not \(14\,\text{g}\). SD: The variation in weight of individual bars. SE: The variation in sample means for \(n = 67\).
Ex. 27.7. \(H_0\): \(\mu = 10\) (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 \(P\) extremely small. For both boys and girls, very strong evidence mean sleep time on weekend less than \(10\,\text{h}\).
Ex. 27.9. 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 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. 28: More details about hypothesis tests
Ex. 28.1. Use \(68\)--\(95\)--\(99.7\) rule and 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. 28.3. Half values in Ex. 28.1. 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. 28.5. \(P\) just larger than \(0.05\); 'slight evidence' to support \(H_1\). \(P\) just smaller than \(0.05\); 'moderate evidence' to support \(H_1\). The difference between \(0.0501\) and \(0.0499\) trivial.
Ex. 28.7. 1. Hypotheses about parameters like \(\mu\). 2. RQ two-tailed. 3. \(36.8052\) is sample mean; hypothesis written before data collected. 4. Hypotheses about parameters; \(36.8052\) is sample mean. These hypotheses are asking to test if sample mean is \(36.8052\); we know it is. 5. Hypothesis written before data collected. 6. Hypotheses about parameters.
Ex. 28.9. 1. Conclusion about pop. mean energy intake. 2. Conclusions never about statistics. 3. Conclusion about pop. mean energy intake. 4. \(P\) is \(0.018\), not less than \(0.018\).
Ex. 28.11. Statements 2 and 4 consistent.
Chap. 29: CIs and tests: mean differences (paired data)
Ex. 29.1. 1. Paired. 2. Paired.
Ex. 29.3. How much longer task takes on the PC for each child.
Ex. 29.7. 1. Analysis: farm. Observation: individual fruits. 2. Pairs have same farm management, soil, etc. 3. Not shown. 4. Not shown. 5. Mean increase in fruit weight from normal to dry (i.e., normal minus dry). 6. \(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d \ne 0\). 7. Sampling distribution not drawn. 8. \(t = -0.205\). 9. \(P\) large; from software, \(P = 0.839\). 10. \(24.79\,\text{g}\) lighter in 2014 to \(20.33\,\text{g}\) higher in 2015. 11. Probably stat. valid; \(n\) just less than \(25\). 12. No evidence (\(t = -0.205\); two-tailed \(P = 0.839\)) of mean increase in weight of squash from dry to normal years (mean change: \(2.230\,\text{g}\) (\(95\)% CI \(-24.8\) to \(20.3\,\text{g}\)), heavier in normal year).
Ex. 29.9. 1. Not shown. 2. How much tastier Broccoli is with dip. 3. \(t = 1.699\); approx. one-tailed \(P\) between \(16\)% and \(2.5\)%; not sure if \(P\) larger than \(0.05\), but likely (\(t\)-score quite a distance from \(z = 1\)). Evidence probably doesn't support \(H_1\). 4. Approx. \(95\)% CI: \(-0.92\) to \(11.22\). 5. Stat. valid.
Ex. 29.11. \(\bar{d} = -424.25\); \(s = 2092.693\); \(\text{s.e.}(\bar{d}) = 467.9404\); \(n = 20\): \(t = -0.90663\). \(P > 0.05\) (actually, \(P = 0.376\)): evidence doesn't support \(H_1\). Approx. \(95\)% CI: $-1360.1) to \(511.6308\). Test may not be stat. valid; histogram of data suggests population might have normal distribution); \(P\) so large probably makes little difference.
Ex. 29.13. 1. Diffs are during minus before: positive diffs means during value is higher. 2. \(\text{s.e.}(\bar{d}) = 3.515018\). 3. \(t = 0.762\); \(P\) is large; no evidence of change; \(-4.35\) to \(9.71\,\text{mins}\).
Ex. 29.15. \(1.65\) to \(4.37\) pounds. Possibly not practically important.
Ex. 29.17. Exact CI: \(1.204\) to \(0.069\,\text{cm}\) further barefoot.
Chap. 30: CIs and tests: comparing two means
Ex. 30.1. How much greater the mean lymphocytes cell diameter is compared to tumour cells.
Ex. 30.3. Normal; mean \(\mu_B - \mu_A\); std dev.: \(2.96\,\text{km}\).h\(-1\).
Ex. 30.5. 1. Parameter: \(\mu_M - \mu_F\) (amount M longer than F, on average). Estimate: \(\bar{x}_M - \bar{x}_F = -0.06\,\text{m}\). 2. Not shown. 3. \(0.0928735\) 4. Not shown. 5. \(-0.25\) to \(0.13\,\text{m}\). 6. \(H_0: \mu_F - \mu_M = 0\); \(H_1: \mu_F - \mu_M \ne 0\). 7. \(t = 0.65\); \(P\) very large. 8. No evidence (\(t = 0.65\); two-tailed \(P > 0.10\)) in sample that mean length of adult gray whales is diff in pop. for F (mean: \(12.70\,\text{m}\); std dev.: \(0.611\,\text{m}\)) and M (mean: \(12.07\,\text{m}\); std dev.: \(0.705\,\text{m}\); \(95\)% CI for the diff: \(-1.26\,\text{m}\) to \(0.246\,\text{m}\)). 9. Yes.
Ex. 30.7. 1. \(\mu_P - \mu_E\), reduction in mean duration for those using echinacea. 2. \(0.2728678\) days; echinacea: \(0.2446822\) days 3. \(0.3665054\). 4. Not shown. 5. \(-0.203\) to \(1.263\) days. 6. \(5.85\) to \(6.83\) days. 7. \(H_0\): \(\mu_P - \mu_E = 0\); \(H_1\): \(\mu_P - \mu_E > 0\) (one-tailed). 8. \(0.3665054\). 9. \(t = 1.47\); one-tailed \(P\) between \(0.025\) and \(0.32\); using \(z\)-tables, \(P\) approx. \(0.074\). 10. Slight evidence of diff. 11. Not given. 12. Yes. 13. Probably not practically important (diff \(0.53\) days).
Ex. 30.9. 1. Amount of DMFT greater in non-industrialised countries, based on upper table having negative mean. 2. \(H_0\): \(\mu_I - \mu_{NI} = 0\); \(H_1\): \(\mu_I - \mu_{NI} \ne 0\). 3. \(11.9\) to \(22.5\), greater for industrialised. 4. Very strong evidence in sample (\(P < 0.001\)) mean annual sugar consumption per person diff for industrialised (mean: \(41.8\,\text{kg}\)/person/year) and non-industrialised (mean: \(24.6\,\text{kg}\)/person/year) countries (\(95\)% CI for the diff \(11.95\) to \(22.54\)). 5. Yes.
Ex. 30.11. 1. \(\mu_Y - \mu_O\): mean amount younger women can lean further forward than older. 2. Boxplot. 3. Not shown. 4. Approx.: \(10.166\) to \(18.834\)oC. Exact CI (Row 2): \(9.10\) to \(19.90\)oC. Different: sample sizes small. 5. One. 6. \(H_0\): \(\mu_Y - \mu_O = 0\); \(H_1\): \(\mu_Y - \mu_O > 0\). 7. \(t = 6.69\) (from second row); \(P < 0.001/2\) as one-tailed; i.e., \(P < 0.0005\). 8. Very strong evidence exists in sample (\(t = 6.691\); one-tailed \(P < 0.0005\)) that pop. mean one-step fall recovery angle for healthy women 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 diff: \(9.1\)oC to \(19.9\)oC). 9. Probably not stat. valid.
Ex. 30.13. \(H_0\): \(\mu_M - \mu_{F} = 0\); \(H_1\): \(\mu_M - \mu_{F} \ne 0\). From output, \(t = -2.285\); (two-tailed) \(P = 0.024\). Moderate evidence (\(P = 0.024\)) mean internal body temperature diff for F (mean: \(36.886\)oC) and M (mean: \(36.725\)oC). Diff between the means (\(0.16\) of degree) of little practical importance.
Ex. 30.15. 1. \(2.76\,\text{kg}\). 2. CB: \(0.227\) to \(5.79\,\text{kg}\); Control: \(-3.68\) to \(2.78\,\text{kg}\). 3. \(-0.68\) to \(7.59\,\text{kg}\), greater for CB. 4. \(t = 1.68\); Two-tailed \(P = 0.100\).
Ex. 30.17. \(t = 2.631\); one-tailed \(P = 0.0055\); evidence of diff.
Chap. 31: CIs and tests: comparing two odds or proportions
Ex. 31.1. Both odds: \(6.04\).
Ex. 31.3. Normal; mean \(p_P - p_N\) and std dev. \(0.0428\). For OR: not normal distribution.
Ex. 31.5. 1. \(z = 3.27\). 2. Very small.
Ex. 31.7. 1. No-PG proportion (Row 1), minus PG proportion (Row 2). 2. \(H_0\): \(p_{\text{With}} - p_{\text{W'out}} = 0\); \(H_1\): \(p_{\text{With}} - p_{\text{W'out}} \ne 0\). 3. \(z = 1.31\); \(P = 0.253\); no evidence of diff. 4. \(-0.071\) to \(0.296\), larger intent to purchase for those without PG study. 5. Odds yes (Col 1), comparing no-PG (Row 1) to PG (Row 2). 6. One option: \(H_0\): \(\text{OR} = 1\); \(H_1\): \(\text{OR}\ne 1\). 7. \(\chi^2 = 1.31\); \(P = 0.253\); no evidence of diff. 8. OR between \(0.68\), \(4.28\). 9. Smallest expected: \(10.6\); yes.
Ex. 31.9. 1. Not shown. 2. \(0.3000\); \(0.3033\). 3. \(-0.0033\). 4. \(\text{s.e.}(\hat{p}_1) = 0.0648074\); \(\text{s.e.}(\hat{p}_2) = 0.0416170\); s.e. for diff.: \(0.077019\). 5. \(-0.0033\pm 0.154\): \(-0.157\) to \(0.151\). 6. \(-0.154\) to \(0.148\). 7. Not given. 8. \(0.429\); \(0.436\). 9. \(0.429/0.436 = 0.985\). 10. \(0.480\) to \(2.02\). 11. Not shown 12. \(p = 0.4333\); s.e. for diff: \(0.0832097\). 13. \(z = (-0.0033 - 0)/0.0832097 = -0.040\); very small \(P\); no evidence of diff. 14. \(\chi^2 = 0.002\) (output); \(P = 0.966\); no evidence of diff.
Ex. 31.11. 1. \(-0.1918\). 2. \(0.04643\). 3. \(0.04202\). 4. Second uses common \(p\): test assumes \(p\) same in both groups. 5. \(-0.1918 \pm (2\times 0.04643)\): \(-0.285\) to \(-0.099\). 6. \(-0.273\) to \(-0.111\). 7. Not given. 8. \(z = (-0.1918 - 0)/0.04202 = -4.56\); small \(P\); evidence of diff. 9. \(0.443\). 10. \(0.315\) to \(0.623\). 11. \(P < 0.0001\); small \(P\); evidence of diff. 12. Yes. 13. Observational.
Ex. 31.13. 1. Not shown. 2. F: \(0.060\), M: \(0.205\); diff: \(0.145\). 3. \(0.0640\), \(0.256\); \(0.250\). 4. s.e. for diff: \(0.0243\); \(0.096\) to \(0.193\). 5. \(0.098\) to \(0.192\). 6. Not given. 7. \(2.45\) to \(6.61\). 8. \(P < 0.0001\). 9. Evidence of a diff. 10. Yes.
Ex. 31.15. 1. OR: \(0.3478261\). 2. Diff: \(-0.12\). 3. Proportions equal; not equal. 4. Odds equal; not equal. 5. \(z = 2.12\); \(P\) 'small'. 6. Some evidence of diff. 7. Yes.
Ex. 31.17. 1. \(H_0\): No association; \(H_1\): Association. 2. \(23.0522\); \(P = 0.00004\). 3. Very strong evidence of association. 4. Yes.
Chap. 32: Finding sample sizes for CIs
Ex. 32.1. Larger.
Ex. 32.3. 1. At least \(25\). 2. At least \(100\) (\(4\) times as many). 3. At least \(400\) (\(16\) times as many). 4. Halve width: \(4\) times as many. 5. Quarter width: \(16\) times as many. 6. More needed for greater precision.
Ex. 32.5. 1. At least \(10\,000\). 2. At least \(2\,500\). 3. At least \(1\,000\). 4. Expensive (time and money): \(10\,000\) and \(2\,500\) probably unrealistic.
Ex. 32.7. Use \(s = 0.43\). 1. At least \(1\,849\). 2. At least \(296\). 3. At least \(74\). 4. Expensive (time and money); \(74\) more realistic.
Ex. 32.9. Use, say, \(s = 13\). 1. At least \(81\) pairs. 2. At least \(76\) pairs.
Ex. 32.11. Use, say, \(s = 0.35\). 1. At least \(44\) in each group. 2. At least \(98\) in each group. 3. Info not relevant to goldfish.
Ex. 32.13. \(2\,223\)
Chap. 33: Correlation and regression
Ex. 33.1. Answers very approximate. 1. \(r\) moderate, positive; \(\hat{y} = 4 + 1.4x\) 2. \(r\) reasonably strong, positive; \(\hat{y} = 6 + 2.3x\). 3. \(r\) not apt: variation in \(y\) increases as \(x\) increases. 4. \(r\) reasonably strong, negative; \(\hat{y} = 8 - 1.5x\).
Ex. 33.3. Any could be.
Ex. 33.5. 1. \(b_0 = 3.5\); \(b_1 = -0.14\). 2. \(b_0 = 2.1\); \(b_1 = -0.0047\). 3. \(b_0 = -25.2\); \(b_1 = -0.95\). 4. \(b_0 = 0.15\); \(b_1 = -0.22\).
Ex. 33.7. Not shown.
Ex. 33.9. 1. \(H_0\): \(\rho = 0\); \(H_1\): \(\rho \ne 0\). 2. No evidence of relationship. 3. Stat. valid if approx. linear; variation in STAI same for diff levels of experience. \(n \ge 25\).
Ex. 33.11. No.
Ex. 33.13. 1. \(b_0\): No time spent on application, mean \(0.27\,\text{g}\) applied; nonsense. \(b_1\): Each extra minute adds average of \(2.21\,\text{g}\) sunscreen. 2. Slope: grams/min; intercept: grams. 3. \(\beta_0\) could be zero; makes sense. 4. \(\hat{y} = 18\,\text{g}\). 5. \(64\)% reduction in unexplained variation using application time. 6. \(r = 0.8\); strong positive correlation.
Ex. 33.15. 1. Probably linear; increasing; approx. constant variance in \(y\) as \(x\) increases. 2. \(H_0\): \(\rho = 0\); \(H_0\): \(\rho > 0\). 3. \(r = 0.837\); \(P < 0.001\). Very strong evidence of positive relationship. 4. Yes.
Ex. 33.17. 1. \(r = 0.264\). 2. \(R^2 = 6.97\)%; using neck circumference reduces unknown variation by about \(7\)%. 3. \(\hat{y} = -24.47 + 1.36x\): \(y\) is REI; \(x\) is neck circum. (in cm). 4. Each \(1\,\text{cm}\) increase in neck circum. increases REI by average of \(1.36\). 5. Approx. CI: \(0.0575\) to \(2.675\). 6. \(t = 2.09\), \(P = 0.041\): slight evidence of relationship. 7. Stat. valid.
Ex. 33.19. 1. Very strong, negative linear relationship. 2. \(r = -\sqrt{0.9929} = -0.9964\); must be negative. 3. \(\hat{y} = 17.47 - 2.59x\): \(x\) is percentage bitumen by wt; \(y\) is percentage air voids by volume. 4. Slope: increase in bitumen wt by one percentage point decreases average percentage air voids by volume by \(2.59\) percentage points. Intercept: extrapolation: \(0\)% bitumen content by wt, percentage air voids by volume \(17.47\)%. 5. \(t = -74.9\): massive; extremely strong evidence (\(P < 0.001\)) of relationship. 6. \(P < 0.001\), as for slope. 7. \(\hat{y} = 4.5027\), or \(4.5\)%; good prediction, as relationship strong. 8. \(\hat{y} = 1.909\), or \(1.9\)%; perhaps poor: extrapolation. 9. Yes.
Ex. 33.21. 1. \(r = 0.271\); \(R^2 = 7.3\)%. 2. \(t = 1.35\); \(P = 0.190\); no evidence of relationship. 3. \(\hat{y} = -5.647 + 0.123x\).
Ex. 33.23. 1. Left probably F. 2. Sex B. 3. A: \(r = 0.600\); B: \(r = 0.815\). 4. B (M). 5. A: \(\hat{y} = -2289 + 21.31x\); B: \(\hat{y} = -3621 + 27.63x\). 6. Both: \(P < 0 .0001\) (probably use one-tailed test). 7. A: \(2\,503\,\text{kg}\); B: \(2\,596\,\text{kg}\). 8. Sample sizes, linearity OK; for Sex B, perhaps hint of increasing variation.
Ex. 33.25. Non-linear relationship.
Chap. 34: Selecting an analysis
Ex. 34.1. 1. Yes. 2. Almost certainly; \(n = 24\) very close to \(n = 25\). 3. No: sample ORs do not have an approx. normal sampling distribution.
Ex. 34.3. Summary of mean diffs; histogram of diffs. Paired samples \(t\)-test; CI for mean diff.
Ex. 34.5. Comparing two odds: odds ratios; stacked, side-by-side bar chart. CI for odds ratio.
Ex. 34.7. Correlation or regression, if linear.
Chap. 35: Writing and reporting research
Ex. 35.1. 1. to. 2. its. 3. One sample with \(50\) individuals; use 'mean' or 'median', not 'average'; units of age is 'years'. 4. Should be one sentence.
Ex. 35.3. 1. Ambiguous; sound like cage is M; passive. '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. 35.5. 1. 'Substantial' if a large change is expected (quote statistics (e.g., \(P\)-value) if statistically significant intended). 2. 'The data are...'
Ex. 35.7. Number decimal places ridiculous.
Ex. 35.9. RQ: P, O, C and I unclear; fonts should be identified. Perhaps better: For students, is mean reading speed for text in Georgia font same as for Calibri font? Abstract statement poor (fonts are not fast or slow). Perhaps: Sample provided evidence mean reading speeds different (\(P = ?\)), when comparing text in Georgia font (mean: ?) and Calibri font (mean: ?; \(95\)% CI for diff: ? to ?).
Ex. 35.11. Variables qualitative: means inappropriate; use odds ratio; values almost certainly refer to CI for OR. Without more information, we can't be sure what the OR means.
Chap. 36: Reading and critiquing research
Ex. 36.1. 1. Convenience; self-selected. Those in study may record different accuracies than those not in. 2. Inclusion criteria. 3. Ethical (drop-outs happen); accurate description of study. 4. Not ecologically valid. 5. Paired \(t\)-test. 6. Null: No mean diff between counts on phone, manually counted; alternative: a diff. 7. \(P\) small; evidence mean diff in step-count between methods cannot be explained by chance: likely a diff. 8. Valid.
Ex. 36.3. 1. Only some evidence of diff in mean age. 2. Comparing the two groups; age possible confounder. 3. Two-sample \(t\)-test. 4. \(0.03376\). 5. \(t = 2.07\); small \(P\); evidence of diff. 6. Probably, given std errors rounded. 7. Conceptual. 8. Not shown. 9. \(\chi^2\). 10. \(z = 1.75\); \(P\) between \(5\)% and \(32\)%: not helpful. 11. Observational: not cause-and-effect; no confounders noted; very restricted population.
Ex. 36.5. 1. \(\chi^2\)-test to compare proportions. 2. No evidence of diff in survival rates at temps. 3. Evidence surviving Cx. had 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. Cx.: evidence mean sizes at temps diff; Ae.: no evidence mean sizes at temps diff. 10. Two-sample \(t\)-tests. 11. Intercept: \(-55.40\) to \(16.28\); slope: \(3.88\) to \(59.40\). 12. \(t = 2.28\); expect small \(P\); evidence of linear association. 13. Need scatterplot to be sure, but \(n \ge 25\). 14. When predator--size ratio increases by one, predation efficiency increases by \(31.64\) percentage points. 15. Unknown variation reduces by \(8.7\)% using predator--size ratio. 16. \(r = 0.294\).
Ex. 36.7. 1. Stratified? 2. Two-sample \(t\)-test. 3. Strong evidence the mean number of actinomycetes diff. 4. Possibly not; sample sizes small. 5. Two-sample \(t\)-test. 6. Very strong evidence mean number higher in CNV. 7. Larger actinomycetes numbers linearly associated with lower corky root severity. 8. \(R^2 = 57.8\)%; unknown variation decreases by \(57.8\)% using actinomycete abundance.