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. 2.2. 1. Percentage people who owns a car. 2. Average time for seedlings to sprout. 3. Average amount of caffeine in soft drinks.
Ex. 2.3. 1. Type of diet. 2. Whether coffee is caffeinated or decaffeinated. 3. Number iron tablets per day.
Ex. 2.4. 1. Whether the vegetables frozen or fresh. 2. Type of fuel. 3. Size of city.
Ex. 2.5. 1. Between-individuals. Outcome: percentage wearing hats. 2. Between-intervals. Outcome: average yield (in kg/plant, tomatoes/plant, etc).
Ex. 2.6. 1. Within-individuals. Outcome: average balance times. 2. Within-individuals comparison. Outcome: average cholesterol levels.
Ex. 2.7. 1. Correlational. 2. No sense assigning one variable as explanatory, another response.
Ex. 2.8. 1. Correlational. 2. Age probably impacts typing speed; age: explanatory; typing speed: 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. 2.10. 1. Some elements not well defined; perhaps: P: Children aged under \(3\) in a Peruvian peri-urban community; O: proportion of children with diarrhoea; C: nutritional status; no intervention. 2. Perhaps: 'In children aged under \(3\) in a Peruvian peri-urban community, is there a relationship between diarrhoea status and nutritional status?'. 3. 'Nutritional status' not well defined. Perhaps amount intake per day? Then, correlational probably. 4. Decision-making. 5. 'Diarrhea status'; 'nutritional status'; probably others. 6. Response: diarrhoea status; explanatory: nutritional status.
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. 2.12. 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. 2.13. 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. 2.14. Outcome should be average lung capacity. Perhaps: 'For students that study at (University), Sippy Downs, do males have a larger average lung capacity than females?'
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. 2.16. False. True. True. Sample size: \(20\).
Ex. 2.17. 1. Analysis: person; observation: individual nose hairs. Each unit of analysis has \(50\) units of observation. 2. \(n = 2\).
Ex. 2.18. 1. Between environments. 2. Before and after for each individual. 3. Operational.
Ex. 2.19. 1. P: American adults; individuals: American adults. 2. O: average number of recorded steps. 3. Response: number of steps recorded for individuals. Explanatory: location of accelerometer. 4. Within individuals.
Ex. 2.20. 1. Descriptive; estimation. 2. Descriptive; decision-making. 3. Correlational; estimation. 4. Relational; decision-making.
Unclear if intervention; seems unlikely.
Ex. 2.21. 1. Relational; decision-making. 2. Correlational; estimation.
Unclear if intervention; seems unlikely.
Ex. 2.22. 1. Relational; estimation. 2. Relational; decision-making. Intervention (whether worms were used or not).
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. 2.24. Analysis: \(12\) subjects. Observation: \(6\) per subjects: \(72\).
Ex. 2.25. The board. Five units of analysis. Ten. Ten. Within-board variation much smaller (apart from first board).
Ex. 2.26.
1. Should be worded terms of averages or means.
2. Everyone dies; a time-frame is probably meant (i.e., within twelve months after amputation).
And compared to what?
3. No defined population (frozen beans? fresh butter beans? tinned kidney beans? jelly beans? coffee beans? can sizes?); "amount" of salt should be, for example, the "average" amount (or concentration) of salt, and per
100 g or similar.
Anyway: You can look at the label (unless, of course, you wish to query those values).
4. Silly: elephants are huge, and joeys are tiny; no-one needs to test this.
Also,the RQ talks about "weight" rather than 'mean weight' too.
5. Needs to be more specific! Perhaps: 'Is the mean reaction time different for males and females?' or similar.
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. 3.2. Response: depression. Explanatory: diet quality. Extraneous: presumably all listed as that is why the researchers obtained this information. None can be lurking variables: researchers measure or observe them.
Confounding variable potentially related to both the response and explanatory variables. Hence, all of the extraneous variables could potentially be confounding variables.
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. 3.4. Older children more likely to be smokers, and would be larger in general: age is a confounding variable. Age is easy to record, and usually is recorded in these types of studies, so probably not a lurking variable.
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. 4.2. 1. Between-individuals. 2. Relational if the C is between individuals. 3. Yes. 4. Decision-making. 5. Experimental. True experiment.
Ex. 4.3. True experiment.
Ex. 4.4. 1. Relational. 2. Between-individuals if it is relational. 3. Yes, since the program were imposed. 4. Experimental. Program given to paramedics in certain cities; quasi-experimental (researchers cannot tell paramedics where to work).
Ex. 4.5. Quasi-experiment.
Ex. 4.6. 1. Many answers possible. 2. Researchers need to intervene: to give or not give subjects a pet. 3. Researchers need to not intervene: find the subjects who already own a pet, or who did not already own a pet.
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. 5: Ethics
Chap. 6: Sampling
Ex. 6.1. c. Externally-valid study more likely.
Ex. 6.2. d. Precise estimates 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. 6.4. Remember: some books borrowed, and not physically in the library! 1. Simple random sample: list of all the books held by the library is needed. May be possible, but probably not for a student or non-library staff member. In principle: number each book, randomly select a sample from that list. 2. Stratified: campuses as strata; a random sample of all the book in each location. 3. Cluster: each shelf as a cluster; randomly select some shelves; determine the number of pages in each book on the selected shelves. 4. Convenience: finding books in the libraries easily accessible. 5. Multi-stage: random sample of campuses; random sample of shelves in the selected libraries; random shelf from each one; a small number of random book from each shelf. 6. Multi-stage perhaps...?
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. 6.6. 1. Convenience; approaching every \(10\)th person may make it a little more representative. 2. Convenience; by approaching every \(5\)th person and going every day for a week, trying to make it a little more representative (and better than the first) 3. Self-selecting. 4. Convenience. By going every day for two weeks, at different times and locations each week, and approaching someone every \(15\) mins, the sample more representative.
The fourth is best, but still far from 'random'. None truly random.
Ex. 6.7. Random sampling to select schools. Then, self-selecting.
Ex. 6.8. True; False; True.
Ex. 6.9. Stratified: zones are strata.
Ex. 6.10. No answer (yet).
Ex. 6.11. In Stage \(2\), selection of farms not random.
Ex. 6.12. In Stage \(1\), selection of schools 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. 7.2. False; true; false; false; true.
Ex. 7.3. Yes; yes; yes; yes; yes; no (external validity).
Ex. 7.3. Yes; yes; yes; no; yes; no (external validity).
Ex. 7.5. Probably in case hive size is a confounder.
Ex. 7.6. Lurking; confounding.
Ex. 7.7. Statements 1, 3, 4, 8 and 9 true. 'Sex', 'Initial weight' possible confounders.
Ex. 7.8. 1.. Observational. 2. PM\(2.5\) concentration; number of smokers. 3. Possibly: wind speed; amount of cover. 4. Yes: be discreet. 5. No.
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. 7.10. 1. Experimental. 2. A group receiving a pill like Treatment A and B, with no effective ingredient. 3. Blinding participants. 4. To ensure participants do not change behaviour because of the treatment received.
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. 7.12. Random allocation: not possible (observational). Blinding: students unaware of which water they drink; in observational study, probably infeasible. Double blinding: neither students nor researchers know which type of water students are drinking; probably infeasible. Control: not sensible. Random sample: any random sampling method preferred; possible, but unlikely.
Ex. 7.13. Carry-over effect; observer effect.
Ex. 7.14. 1. Random allocation; blocking; recording potential confounders. 2. Blinding participants, researchers. 3. Change in nasal congestion. 4. Type of cleaning. 5. Age; sex.
Ex. 7.15. No; also possible in observational studies.
Chap. 8: Research design limitations
Ex. 8.1. External.
Ex. 8.2. Ecological; external; internal; confounding; sampling.
Ex. 8.3. Population: 'on-campus university students where (I) work'. External validity: whether the results apply to other members of target population.
Ex. 8.4. Not ecologically valid (e.g., parachutes used at high altitude). Probably not externally valid: voluntary (and presumably convenience) sample. Internal validity may be fine...
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. 8.6. Internal.
Ex. 8.7. Observational study: people with severe cough may take more cough drops.
Ex. 8.8.
1. Study is observational because the researchers cannot determine the comparison (not the outcome). 2. This is a mix of both C ('smokers and non-smokers') and O ('the median serum cholesterol'). 3. External validity only refers to whether the sample represents the given target population, which is Australians. Whether the results apply for the entire world is irrelevant. 4. 'Serum cholesterol' is not a variable; nothing here is varying. 'Serum cholesterol' is just a type of cholesterol. The variable is 'the serum cholesterol concentration' or similar. 5. This is not an experiment. 6. The data file will have two columns (variables), but one column will record the smoking status (Yes/No) and one column will record the serum cholesterol concentration. 7. A confounding variable has to be related to both the response and explanatory variables. 8. The observer effect is about how the researchers might respond, not the individuals under study.
Ex. 8.9. Patients probably knew they were involved; Hawthorne effect should be considered in interpretation.
Chap. 9: Collecting data
Ex. 9.1. No place for \(18\)-year-olds.
Ex. 9.2. '\(2\) children' belong in two categories (not mutually exclusive); '\(4\) children' does not belong in any category (not exhaustive).
Ex. 9.3. Best: second. First: leading (concerned cat owners...) Third: leading (Do you agree...)
Ex. 9.4. First: leading ('environmentally friendly'). First and second: only present the option of 'owning' Third: best question.
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).
Ex. 9.6. The first two are closed, with one option to be selected. The other is open. I'm not sure primary school children could recall the answers to the first two questions accurately...
Chap. 10: Classifying data
Ex. 10.1. Quant. continuous. Qual. nominal. Quant. continuous.
Ex. 10.2. Quant. discrete. Qual. nominal. Qual. nominal.
Ex. 10.3. False; true; false
Ex. 10.4. False; false; true.
Ex. 10.5. Nominal; qualitative.
Ex. 10.6. Foliage biomass: quant. continuous. Tree diameter (in cm): quant. continuous. Age of the tree (in years): quant. continuous. Origin of the tree: qual. nominal.
Ex. 10.7. 1. Blood pressure: quant. continuous. 2. Program: qual. nominal. 3. Grade: qual. ordinal. 4. Number of doctor visits: quant. discrete.
Ex. 10.8. 1. Age: qual. ordinal. 2. Gender: qual. nominal. 3. Location: qual. nominal. 4. Social media use: qual. ordinal. 5. BMI: quant. continuous. 6. Total sitting time, in minutes per day: quant. continuous.
Ex. 10.9. Gender: qual. nominal. Age: quant. continuous. Height: quant. continuous. Weight: quant. continuous. GMFCS: qual. ordinal.
Ex. 10.10. Fertilizer dose: quant. continuous. Soil nitrogen: quant. continuous. Fertilizer source: qual. nominal.
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.
Ex. 10.12. Location is only variable (observed from the individuals). Number of people and percentage of people who died is a summary of the data collected from individuals. 'Location' is a nominal, qualitative variable; seven levels.
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. 11.2. Very right-skewed. Average somewhere near \(1\) or \(2\) k perhaps. Most between \(0\) and about \(5\) kg. Possible outlier near \(11\) kg.
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. 11.4. Average around \(15.0\) g. Most between \(13.5\) and \(16.0\) kg. Slightly skewed left, with perhaps one low outlier. Maybe quote both the mean and median (\(14.90\) and \(14.99\) respectively).
Ex. 11.5. 1. \(3.7\). 2. \(3.5\). 3. \(1.888562\).
Ex. 11.6. 2. \(643\) g. 3. \(12.884\) g. 4. \(637.5\) g. 5. We do not know.
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. 11.8. 1. Fig. not shown. 2. \(\bar{x} = 144.3\), g; median: \(143.3\) g; \(s = 11.61\), g; IQR: \(16.25\) g. 3. Unlikely (but is only one of many possible samples).
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. 11.10. 2. \(\bar{x} = 3.09\); median: \(2.0\). 3. \(s = 2.77\). 4. IQR: \(4\).
Ex. 11.10. Average: hard to be sure... maybe between \(10\) or \(15\). Variation: about \(0\) to about \(40\). Shape: slightly skewed right. Outliers: no outliers or unusual observations; the observation between \(35\) and \(40\) may be an outlier. I suspect it is not; a larger sample may very well have observations between \(30\) and \(35\). I could be wrong.
Ex. 11.11. No answer (yet).
Ex. 11.12. Dataset B: more observations close to the mean; the average distance from mean would be smaller. The standard deviation for Dataset B smaller than for Dataset A.
Ex. 11.13. D; C; A; D.
Ex. 11.14. D; C; B; D; C.
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. 12.2. 1. Where: nominal (respondents can select more than one option; each option effectively a Yes/No variable). Temp: quantitative, but three options: can treat as ordinal. Time: ordinal. 2. Graphs not shown. 3. Where: home, work most common (modes). Temp: \(100^\circ\)C is mode and median among those who know. Time: \(3\) mins is mode and median among those who know. Counts can be turned to percentages and odds too.
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. 12.4. None are bad. I'd prefer bar chart, but any OK.
Ex. 12.5. Bar (or dot) chart. Pie chart inappropriate: more than one option can be selected.
Ex. 12.6. 1. Nominal: gender; ordinal: place of residence; responses. 2. Gender: modes are F and M. Place: City \(> 100\ 000\) residents. Response: Agree. 3. Gender: NA. Place: City \(20\ 000\) to \(100\ 000\) residents. Response: Neutral. 4. If the response are considered equally spaced, assign \(1\) to 'Strongly agree' up to \(5\) to 'Strongly disagree'; then mean is \(3.02\), effectively 'Neutral'.
Ex. 12.7. Plots not shown. 4. Advantage: availability. Disadvantage: high price. 5. Table not shown. 6. \(0.209\). 7. \(1.21\).
Ex. 12.8. 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. \(39.1%\). \(28.7\)%. \(0.642\). \(0.402\). \(1.6\). Table not shown.
Ex. 12.10. 1. Type of student: nominal (three levels). Number of concussions: ordinal (three levels). 2. \(18.75\)% of all college students in the sample have received one concussion. 3. Odds(non-athlete receiving \(2+\) concussions): \(0.35\). Non-athletes \(0.35\) times as likely to receive two or more concussions than fewer than two. Or: For every \(100\) students with fewer than two concussions, there are \(35\) with two or more. 4. Odds(soccer player receiving \(2+\) concussions): \(0.26\). 5. OR: About \(1.35\). 6. Not given. 7. Not given. 8.. Probably row percentages.
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. 13.2. 1. \(0.32616\). 2. About \(2.07\). 3. About \(1.69\). 4. About \(37.1\)%. 5. \(1.22\).
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. 13.4. 1. \(45.9\)%. 2. \(61.4\)%. 3. \(0.848\). 4. \(1.59\). 5. \(1.15\). 6. \(0.533\). 7. The odds of reporting back pain from carrying school bags, comparing boys to girls.
Ex. 13.5. Plot not shown.
Ex. 13.6. 1. Season; sex. 2. Figure not shown. 3. Most cannot be sexed! Lower numbers in spring and summer
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. 13.8. 1. \(13.2\)%. 2. \(2.3\)%. 3. \(0.152\). 4. \(0.0238\). 5. \(6.39\). 6. Odds coffee drinker being a smoker is \(6.39\) times the odds of a non-coffee drinker being smoker. 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. 15.2. In general, female basketballers taller than female netballers (first, second and third quartiles all greater for basketballers). Second and third quartiles, differences quite substantial. The minimum heights are similar.
Ex. 15.3. A. II (median; IQR). B. I (mean; standard deviation). C. III (median; IQR).
Ex. 15.4. Worker 2 faster in general (more panels installed per minute), including one fast outlier. Workers 1 and 3 similar medians; Worker 3 more consistent (smaller IQR).
Ex. 15.5. Plot not shown.
Ex. 15.6. Figures not shown.
Ex. 15.7. No answer (yet).
Ex. 15.9. 1. Table not shown. 2. Plot not shown.
Ex. 15.10. No answer (yet).
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. 15.12. 1. Neither correct: smallest jellyfish has \(6\)cm breadth. 2. The sample is small, so hard to be certain. Average: about \(10\)mm; data from about \(6\) to \(16\)mm; slightly right skewed; no outliers. 3. Site B generally larger. 4. Site A.
Chap. 16: Quantitative data: Correlations between individuals
Ex. 16.1. Many correct answers.
Ex. 16.2. You cannot be precise with guesses. A: Large; positive. B: Moderate; negative. C: Near zero. D: Not appropriate.
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. 16.4. Correlation coefficients very hard to estimate! From software: \(r = -0.13\). Realistically, the best you can do is 'slightly negative'.
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. 16.6. Approximately linear; positive relationship; variation seems to get larger for a larger number of cases.
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. 16.8. No answer (yet).
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. 16.10. No answer (yet).
Ex. 16.11. No answer (yet).
Ex. 16.12. \(R^2 = (-0.682)^2 = 0.465\): 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. 17.2. Individual variables: bar chart for origin; histogram for others. Relationships are main focus. Between biomass, origin: boxplot. Between biomass, other variables: scatterplot. (On scatterplot, origins could be encoded with different colours or symbols.)
Ex. 17.3. Not shown.
Ex. 17.4. Fertilizer (quant.): histogram (response). Soil nitrogen (quant.): histogram (explanatory). Source (qual. nominal): bar chart (explanatory). Relationships: between fertilizer dose, soil nitrogen: scatterplot. Source could be encoded using different coloured points.
Ex. 17.5. Plotting symbols unexplained. Axis labels unhelpful. Vertical axis could stop at \(20\).
Ex. 17.6. Graph inappropriate: both variables qualitative. Use stacked or side-by-side bar chart.
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. 17.8. Variable is 'Sport' (qual. nominal). The bars can be ordered any way. Skewness makes no sense: only sensible for quant. variables.
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?
Ex. 17.10. Graph: odd colour choice; vertical axis label unhelpful; horizontal axis unlabelled; units of measurement not given; title and/or caption helpful.
Table: CI limits under the Mean and Std dev columns; units of measurement not given; no caption, or explanation of table; number of decimal places inconsistent; sample sizes not given; difference (and prob. other rows) should report a std. error.
Chap. 18: Probability
Ex. 18.1. 1. Subjective. 2. Rel. frequency.
Ex. 18.2. 1. Classical. 2. Probably subjective.
Ex. 18.3. False. True. \(1/2\). \(1/2\).
Ex. 18.4.
Only 
.
\(1/6\).
All outcomes; this represents the whole sample space.
\(1\).
Only .
\(1/6\).
, 
, 
, 
and 
.
\(5/6\).
\(1/3\).
\(0\).
Ex. 18.5. \(4/6\). \(5\). Yes: what happens on die won't change coin outcome. \(1/2\). \(1/6\). \(1/3\).
Ex. 18.6. \(4/52 = 0.07692\). \(4/48 = 0.08333\). \(16/52 = 0.3077\). \(16/(52 - 16) = 0.4444\). Not independent (like Example 18.12). Are independent; what happens on the die does not change what happens with the cards. \(4/16 = 0.25\).
Ex. 18.7. No answer (yet).
Ex. 18.8. 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. 18.9. 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. 18.10. 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.
Ex. 18.11. Events not equally likely.
Ex. 18.12. 1. \(49.2\)%. 2. \(17.7\)%. 3. \(55.1\)%.
Chap. 19: Making decisions
Ex. 19.1. 1. Yes! Problem seems likely (can't be sure). 2. Assuming fair die, would not expect ⚅ ten times in a row.
Ex. 19.2. 1. That the population mean is \(12\) inches, as claimed. We have no evidence to refute this. 2. First: population mean diameter is \(\mu = 12\) inches; the sample mean is not \(12\) inches due to sampling variation. Second: population mean diameter isn't \(12\) inches, reflected in the sample. 3. \(11.48\) is \(0.52\) inches from target of \(12\); seems unlikely the sample mean would be that far from \(12\) inches through sampling variation alone. 4. \(\bar{x} = 11.25\) inches is further from \(\mu = 12\) that \(\bar{x} = 11.48\): claim probably not supported. 5. Smaller sample sizes: sample mean would vary more (in general, larger samples give more precise estimates).
Ex. 19.3. Seems unlikely.
Chap. 20: Sampling variation
Chap. 21: Distributions and models
Ex. 21.1. All false.
Ex. 21.2. All false.
Ex. 21.3. 1: C; 2: A; 3: B; 4: D.
Ex. 21.4. 1: A; 2: C; 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. 21.6. 1. About \(3.1\)%. 2. About \(3.8\)%. 3. About \(48.7\)%. 4. Smaller than about \(7.38\) mm. 5. Larger than about \(7.21\) mm.
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. 21.8. \(z\)-score: about \(z = 2.05\). Corresponding IQ: \(130.75\). An IQ greater than about \(130\) is required to join Mensa.
Ex. 21.9. Lower than about \(80.8\): rejection.
Ex. 21.10. Be very careful: work with number of minutes from the mean, or from 5:30pm. The std. deviation \(120\) mis, plus \(0.28\times 60 = 16.8\) mins. The std. deviation is \(136.8\) mins.
1. \(9\)pm is \(3\) h and \(30\) mins from \(5\):\(30\)pm: \(210\) mins. \(z\)-score: \(z = 1.54\); probability: about \(6.2\)%. 2. \(z = -0.22\); probability: \(41.3\)%. 3. \(z_1 = -0.22\) and \(z_2 = 0.22\); probability: \(0.5871 - 0.4129\); about \(17.4\)%. 4. \(z\)-score is \(0.52\); time is \(x = 71.136\) minutes after \(5\)pm; about one hour and \(11\) mins after \(5\):\(30\)pm, or \(6\):\(41\)pm. 5. \(z\)-score: \(-1.04\); time is \(x = -141.272\), or \(141.272\) mins before \(5\)pm; about two hours and \(21\) mins before \(5\):\(30\)pm, or \(3\):\(09\)pm.
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. 23.2. \(\hat{p} = 791/1766 = 0.4479049\) and \(n = 1766\), \(\text{s.e.}(\hat{p}) = 0.01183326\). The approx. \(95\)% CI from \(0.424\) to \(0.472\). (Many decimal places were kept in working' final answers rounded.)
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. 23.4. \(\hat{p} = 18/51 = 0.3529\); \(\text{s.e.}(\hat{p}) = 0.06692\). Approx. \(95\)% CI: \(0.3529 \pm 0.1338\). Margin of error: \(0.1338\). Based on the sample, the sample proportion of koalas that crossed at least one road in the previous \(30\) months is \(0.353\) (\(n = 51\)), with an approximate \(95\)% CI from \(0.219\) to \(0.487\).
The number of koalas that had crossed a road at least one is \(18\), and the number that had not crossed a road at least once is \(51 - 18 = 33\). Both exceed \(5\), so the CI is 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. 23.6. After \(3000\) h: \(\hat{p} = 0.2143\); \(\text{s.e.}(\hat{p}) = 0.06331\). CI: from \(0.088\) to \(0.341\). Stat. valid.
After \(400\) h: \(\hat{p} = 0\); \(\text{s.e.}(\hat{p}) = 0\). CI: \(0\) to \(0\): silly (implies no sampling variation). Because stat. validity conditions not satisfied.
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: \(n > 25\).
Ex. 24.2. 1. The population mean weight of school bags for Iranian children in Grades \(6\)--\(8\). 2. \(\text{s.e.}(\bar{x}) = 0.03883\). 3. CI: \(2.8\pm0.07766\). 4. The sample mean weight of school bags is \(2.8\) kg (s.e.: \(0.0388\); \(n = 586\)), with an approx. \(95\)% CI from \(2.72\) to \(2.88\). 5. Since \(n > 25\); CI is stat. valid.
Ex. 24.3. \(\text{s.e.} = 0.06410062\). Approx. \(95\)% CI: \(2.72\) L to \(2.98\) L.
Ex. 24.4. Std. error: \(\text{s.e.} = 994.2182\) (keeping lots of decimal places in working). Approx. \(95\)% CI: \(4967.984\)\(\mu\)g to \(8944.86\) \(\mu\)g.
Ex. 24.5. Approx. \(95\)% CI : \(29.9\) s to \(36.1\) s.
Ex. 24.6. 1. Std. error: \(\text{s.e.}(\bar{x}) = 46.15526\); approx. \(95\)% CI: \(754.1\) ml to \(938.7\) ml. 2. Don't seem very good at estimating (the article reports guesses ranged from \(50\) ml to \(3000\) ml). 3. Sample size much larger than \(25\); the CI is stat. valid. 4. Using the margin-of-error as \(50\), and \(s = 651.1\): need about \(679\) participants (round up). 5. Using margin-of-error as \(25\), and \(s = 651.1\): need about \(2714\) participants (round up). 6. To halve width of the margin of error, four times as many subjects are needed.
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. 24.8. Neither correct. To learn about the variation in individuals trees, use the std. deviation rather than std. error. The std. error tells us about the population mean diameter, not individual trees.
Ex. 24.9. \(\text{s.e.}(\bar{x}) = 5.36768\); approx. \(95\)% CI: \(50.56\) s to \(72.04\) s. Stat. valid.
Ex. 24.10. 1. \(\mu\): population mean diameter of all EB pizzas; \(\bar{x}\): mean diameter of the pizzas in the sample. 2. \(\bar{x} = 11.486\) inches (not sensible to quote the diameter to \(0.001\) of a cm). Value of \(\mu\) unknown; caimed as \(\mu = 12\). 3. \(s = 0.2465\) inches; \(\sigma\) is the unknown std deviation of the population. 4. \(0.02205\). 5. \(s\): variation in the diameters of individual pizzas; s.e.: the precision of the sample mean when used to estimate the population mean. 6. Almost certainly not the same; probably close to \(11.486\) inches. 7. Normal; mean \(\mu\); std. dev is the standard error of \(0.02205\). 8. Approx. \(95\)% CI: from \(11.44\) to \(11.53\) inches. 9. Not given. 10. \(n \ge 25\), or population has normal distribution. 11. We do not need to assume that \(n > 25\); it is. 12. \(\mu\) probably not \(12\) inches based on the CI.
Chap. 25: More about CIs
Ex. 25.1. CIs give intervals for unknown parameters. (The CI is \(68\)% anyway, not \(95\)%.)
Ex. 25.2. CI not about individual trees; about a population parameter. Presumably: 'This CI means that between \(22.3\)% and \(40.5\)% of trees are infected with apple scab'.
Ex. 25.3. CIs gives interval for population mean, not data.
Ex. 25.4. CI gives an interval for the population mean, and not sample mean.
Chap. 26: CIs for paired data
Ex. 26.1. 1. Paired. 2. Paired.
Ex. 26.2. 1. Paired. 2. Not 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. 26.4.
1. Computing differences as Before minus the After measurements seems sensible: the average blood pressure decrease, the purpose of the drug.
2. The diffs (when defined as reductions): \(9\), \(4\), \(21\), \(3\), \(20\), \(31\), \(17\), \(26\), etc..
3. Mean diff.: \(18.933\); std. deviation: \(9.027\); std. error: \(2.331\).
Approximate \(95\)% CI: \(14.271\) to \(23.56\) mm Hg.
4. Exact \(95\)% CI: \(13.934\) to \(23.93\) mm Hg from output.
5. First uses approx. multipliers; second uses exact multipliers.
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. 26.6. 1. Approx. \(95\)% CI for reduction: \(-0.08\) to \(1.4\): average could be an increase of up to \(0.08\) to a reduction of up to \(1.4\) on the given scale for women. 2. Sample size not larger than \(25\), but close: probably reasonably stat. valid.
Ex. 26.7. No answer (yet).
Ex. 26.8. \(\bar{d} = 7.70\) fmol/mol; std. error for this increase is \(3.10\) fmol/mol. The approx. \(95\)% CI is \(1.50\) to \(13.90\) fmol/m.
Using jamovi: exact \(95\)% CI from \(1.18\) to \(14.22\) fmol/mol, similar to approx. CI computed manually. Sample size is \(n = 19\), just less than \(25\); results may not be stat. valid (but shouldn't be too bad).
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.'
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. 27.2. 1. Table not shown. 2. From jamovi, exact \(95\)% CI: \(0.05438\) to \(0.11501\) (bottom row). Exact \(95\)% CI for the diff. between the mean direct HDL cholesterol concentrations: \(0.05438\) to \(0.11501\) mm Hg higher for non-smokers.
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. 27.4. 1. Exercise: \(0.4427\); splinting: \(0.3479\). 2. \(0.563\) 3. Splinting minus exercise: the diffs are how much greater the pain is with splinting. 4. \(-0.826\) to \(1.426\): \(0.826\) greater pain with exercise to \(1.426\) greater pain with splinting. 5. \(0.404\) to \(1.796\). 6. Sample sizes small; CIs may not be stat. valid; roughly correct only.
Ex. 27.5. No answer (yet).
Ex. 27.6 1. Perhaps: \(\mu_{\text{After}} - \mu_{\text{Before}}\), the increase in deceleration. 2. Approx. CI: \(-0.00162\) to \(0.00562\) m/s. The diff between mean decelerations likely to be between \(-0.0016\) m/s (i.e, a mean acceleration of \(0.0016\) m/s) to \(0.0056\) m/s.
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. 27.8. No answer (yet).
Ex. 27.9. No answer (yet).
Ex. 27.10. No answer (yet).
Ex. 27.11. No answer (yet).
Chap. 28: CIs for odds ratios
Ex. 28.1. One.
Ex. 28.2. Negative.
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. 28.4.
1. \(0.65257\).
2. \(0.61491\).
3. \(0.03766\), as in output.
4. \(\text{s.e.}(\hat{p}_M) = 0.026475\); \(\text{s.e.}(\hat{p}_W) = 0.037526\); \(\text{s.e.}(\hat{p}_M - \hat{p}_W) = 0.04608\);
5. \(-0.0545\) to \(0.130\).
6. \(-0.0533\) to \(0.1287\).
7. A \(95\)% chance that the interval \(-0.0533\) to \(0.1287\) straddles value of the population proportion.
8. About \(1.878\).
9. About \(1.597\).
10. \(1.1763\), as in output.
11. A few ways; for example: For every \(100\) women with a smooth scar, about \(118\) men with a smooth scar.
12. \(0.7966\) to \(1.7370\).
13. Table not shown.
14. (Graph not shown; use a stacked or side-by-side bar chart.)
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. 28.6. 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 \(3000\) h. 4. \(-0.2597\) to \(0.0229\), larger when run for \(3000\) h. 5. No answer (yet). 6. \(0.1331\) to \(1.1366\). 7. Not answer (yet). 8. 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. 28.8.
1. \(0.0975\) to \(0.1916\). 2. \(2.4480\) to \(6.6135\). 3. No answer (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.
Ex. 28.10. 1. Not shown. 2. Not shown. 3. \(1.08\) to \(9.24\). 4. \(-0.0078\) to \(03006\). 5. One expected count just less than five; shouldn't be too concerned (but it should be noted).
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. 29.2. 1. At least \(32\) in each. 2. At least \(128\) (i.e., four times as many). 3. At least \(512\) (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. 29.4. 1. At least \(n = 1/(0.05^2) = 400\). 2. At least \(n = 1/(0.025^2) = 1600\). 3. To halve width, four times as many people needed.
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. 29.6. No answer (yet).
Ex. 29.7. Use, say, \(s = 13\). 1. At least \(81\) pairs. 2. At least \(76\) pairs.
Ex. 29.8. 1. At least \(81\). 2. At least \(144\).
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.
Ex. 29.10. 1. At least \(13\) in each group. 2. At least \(50\) in each group.
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. 30.2. 1. \(\hat{p} = 55/972 = 0.056584...\). 2. \(H_0\): \(p = 0.10\) (i.e, assume target is met, and the diff. between \(p\) and \(\hat{p}\) due to sampling variation); and \(H_1\): \(p < 0.10\). One-tailed: RQ is whether the target is \(10\)% or lower. 3. \(\text{s.e.}(\hat{p}) = 0.0096225...\). 4. \(z = -4.51\); very large (and negative) \(z\)-score, so \(P\)-value very small using \(68\)--\(95\)--\(99.7\) rule or tables. 5. Very strong evidence exists in the sample (\(z = -4.51\); one-tailed \(P < 0.001\)) that the population proportion is less than the target of \(p = 0.10\) (Korean sample proportion: \(\hat{p} = 0.0566\); \(n = 972\); approx. \(95\)% CI from \(0.042\) to \(0.071\)). 6. Stat. validity: \(n\times p = 97.2\) and \(n\times (1 - p) = 874.8\); both exceed \(5\).
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. 30.4. \(H_0\): \(p = 0.5\) and \(H_1\): \(p > 0.5\) (which is one-tailed). \(\hat{p} =0.5759\) and \(n = 158\). \(\text{s.e.}(\hat{p}) = 0.03977786\), so \(z = 1.91\). Using the normal-distribution tables, the one-tailed \(P\)-value is \(0.0281\); slight evidence in the sample of a home-ground advantage in the population.
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. 30.6. \(H_0\): \(p = 0.255\) and \(H_1\): \(p \ne 0.255\). \(\hat{p} = 0.2464986\) and \(n = 357\), so \(\text{s.e.}(\hat{p}) = 0.02306822\), so \(z = -0.368533\). Very small; \(P\)-value large. No evidence exists that the proportion of smokers is different for casino-goers compared to the general U.S. 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. 30.8. No answer (yet).
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. 30.10. \(H_0\): \(p = 1/16 = 0.625\); \(H_1\): \(p \ne 0.0625\). With \(n = 172\), find \(\hat{p} = 0.1395349\) and \(\text{s.e.}(\hat{p}) = 0.01845701\) so \(z = 26.3\), which is massive; the \(P\)-value incredibly small. Very strong evidence that the pop. proportion is not \(1/16\), and the borers were not resistant.
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. 31.2. \(H_0\): \(\mu = 120\) and \(H_1\): \(\mu \ne 120\) (two-tailed), where \(\mu\) is the mean time in seconds. Std. error: \(\text{s.e.}(\bar{x}) = 2.581472\). \(t\)-score: \(-23.13\), which is huge; \(P\)-value really small.
Very strong evidence (\(P < 0.001\)) that children do not spend \(2\) mins (on average) brushing their teeth (mean: \(60.3\) s; std. dev.: \(23.8\) s). 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. 31.4. \(t = -8.5\); very small \(P\)-value. Evidence mean water temp below \(60^\circ\)C.
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. 31.6. \(H_0\): \(\mu = 1000\) and \(H_1\): \(\mu \ne 1000\) (\(\mu\) is the pop. mean guess of spill volume). Std. error: \(46.15526\). \(t = -3.33\): \(P\)-value very small. Very strong evidence that the mean guess of blood volume is not \(1000\) ml, the actual value. The sample is much larger than \(25\): the test is stat. valid.
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. 31.8. Hypotheses have the form \(H_0\): \(\mu = \text{pre-determined target}\), and \(H_1\): \(\mu \ne \text{pre-determined target}\). \(t\)-scores: \(t_1 = 0.318\), \(t_2 = 2.347\), \(t_3 = -0.466\), \(t_4 = -0.726\). \(P\)-values large, except for second test. No evidence that the instruments are dodgy, except perhaps for the first instrument for mid-level LH concentrations. Should be stat. valid. While assessing the means is useful, how variable the measurements are is also useful (but beyond us).
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. 32.2. Using the \(68\)--\(95\)--\(99.7\) rule and a diagram: 1. Bit smaller than \(0.32\) (\(68\)% between \(-1\) and \(1\)). 2. Large-ish: Between \(0.32\) (\(68\)% between \(1\) and \(-1\)) and \(0.05\)% (\(95\)% between \(-2\) and \(2\)), but closer to \(0.32\). 3. Very small; much smaller than \(0.003\). 4. Very large! Almost \(0.50\).
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. 32.4. Half the values in Ex. 32.2. 1. Bit smaller than \(0.16\) (\(68\)% between \(-1\) and \(1\)). 2. Between \(0.16\) (\(68\)% between \(1\) and \(-1\)) and \(0.025\) (\(95\)% between \(-2\) and \(2\)), but closer to \(0.32\). 3. Very small; much smaller than \(0.0015\). 4. Very large! Almost \(0.25\).
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. 32.6. The \(P\)-value just larger than \(0.05\); 'moderate evidence' to support \(H_1\). The \(P\)-value just smaller than \(0.05\); 'strong evidence' to support \(H_1\). The difference between \(0.011\) and \(0.009\) 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. 32.8. 1. Hypotheses about parameters like \(\mu\), not statistics like \(\bar{x}\). \(36.8052\) is a sample mean, but hypothesis are meant to be written before data collected. In any case, these hypotheses are asking to test if the sample mean is \(36.8052\)... which we know it is. 2. \(36.8052\) is sample mean, but hypothesis are meant to be written down before the data are collected. 3. Hypotheses are about parameters like \(\mu\), not statistics like \(\bar{x}\).
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. 32.10. 1. No evidence of a diff. in lifetime between two brands, as the \(P\)-value is 'large'. 2. No; diff. is \(0.29\) h, or about \(17\) mins. A diff. of \(17\) mins in over \(5\) h of use is trivial. 3. Conclusion: no evidence of a diff. between the mean lifetimes; cumbersome for advertising! A common advertising trick: 'No other battery lasts longer!'... meaning there is no evidence of a difference in means. 4. Price!
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. 33.2. How much larger the pH is downstream compared the upstream for each river.
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. 33.4. 1. Because it is the blood pressure reduction, and a reduction is what the drug is meant to produce, so expect the reductions to be positive numbers. 2. Diffs not shown. 3. Histogram of diffs not shown. 4. \(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d > 0\) (because the diffs are reductions). 5. \(t = 8.12\). 6. \(P = 0.001\div 2 = 0.0005\) (one-tailed). 7. Very strong evidence (\(P = 0.0005\)) that the drug reduces the average systolic blood pressure (mean reduction: \(8.6\) mm Hg) in the population.
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. 33.6. \(H_0\): \(\mu_d = 0\) and \(H_1\): \(\mu_d > 0\), where diffs. are positive when the intention to smoke is reduced after exercise. \(t = 1.78\); \(P\)-value larger than \(0.05\): the evidence doesn't support the alternative hypothesis. No evidence (\(P > 0.05\)) the mean 'intention to smoke' reduced after exercise in women (mean change in intention to smoke: \(-0.66\); std. error: \(0.37\)).
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.
Ex. 33.8. \(H_0\): \(\mu_D = 0\) and \(H_1\): \(\mu_D > 0\), if diff. is the beta-endorphin concentration \(10\) mins before surgery, minus the concentration \(12\)--\(14\) hours before. \(t = 2.48\); quite large. Using the \(68\)--\(95\)--\(99.7\) rule, expect small one-tailed \(P\)-value (certainly less than \(0.025\)). jamovi reports one-tailed \(P = 0.0115\). Conclude (where the CI was found earlier) that strong evidence exists in the sample (paired \(t = 2.48\); one-tailed \(P = 0.0115\)) of a population mean increase in beta-endorphin concentrations from \(12\)--\(14\) hours before surgery to \(10\) mins before surgery (\(95\)% CI: \(1.62\) to \(13.78\) fmol/mol.) \(n = 19\), just less than \(25\); results may not be stat. valid (shouldn't be too bad).
Chap. 34: Tests for two means
Ex. 34.1. How much greater the mean lymphocytes cell diameter is compared to tumour cells.
Ex. 34.2. How much greater the mean braking distance is for cars with Type B brake pads compared to Type A brake pads.
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. 34.4. \(H_0\): \(\mu_S - \mu_{NS} = 0\) and \(H_1\): \(\mu_S - \mu_{NS} \ne 0\). From output: \(t = 5.478\) and \(P < 0.001\). Very strong evidence to support \(H_1\).
Ex. 34.5. No answer (yet).
Ex. 34.6. 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. 34.8. No answer (yet).
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. 34.10. 1. \(H_0\): \(\mu_D - \mu_N = 0\) (no diff. in the pop. mean BHADP scores). \(H_1\): \(\mu_D - \mu_N \ne 0\) (diff. in the pop. mean BHADP scores). 2. \(t = \frac{6.76 - 0}{0.78794} = 8.58\). 3. \(P\) will be very small. 4. Strong evidence exists in the sample (\(t = 8.58\); two-tailed \(P < 0.001\)) that people with a disability (mean: \(31.83\); \(n = 132\); standard deviation: \(7.73\)) and people without a disability (mean: \(25.07\); \(n = 137\); standard deviation: \(4.80\)) have different population mean access to health promotion services (\(95\)% CI for the difference: \(5.18\) to \(8.34\)), as measured by BHADP scores.
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. 34.12. 1. Table not given. 2. \(H_0\): \(\mu_C - \mu_{SO} = 0\) and \(H_1\): \(\mu_C - \mu_{SO} \ne 0\); \(t = -1.513\); \(P\)-value larger than \(5\)%. Sample size are small; test may not be stat. valid. 3. \(H_0\): \(\mu_C - \mu_{SO} = 0\) and \(H_1\): \(\mu_C - \mu_{SO} \ne 0\); \(t = -2.70\); \(P\)-value smaller than \(5\)%. Sample size small; may not be stat. valid.
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. 35.1. Both odds: \(0.977\).
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. 35.4. One option: \(H_0\): The pop. OR one; \(H_1\): The pop. OR not one. From software, \(\chi^2 = 0.667\); \(P = 0.414\), which is large. No evidence (\(P = 0.414\)) odds of having a smooth scar is different for women and men (chi-square: \(0.667\)). The test is stat. valid.
Ex. 35.5. No answer (yet).
Ex. 35.6. One option: \(H_0\): The pop. OR is one; \(H_1\): The pop. OR not one. From software, \(\chi^2 = 3.845\); \(P = 0.050\). Moderate evidence (\(P = 0.05\)) odds of having no rainfall is different for non-positive SOI Augusts and negative-SOI Augusts (chi-square: \(3.845\)). The test is stat. valid.
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. 35.8. 1. Not shown. 2. BF: \(0.08696\); control: \(0.25\). 3. BF: \(0.08\); control: \(0.20\). 4. OR: \(0.348\); diff: \(-0.12\). 5. \(H_0\): No association; \(H_1\): association. 6. No given. 7. \(2.12\). 8. Some evidence of association.
Ex. 35.9. From software: \(\chi^2 = 22.374\), approx. \(z = 4.730\): very large; small \(P\)-value. From software: \(P < 0.001\). The sample provides very strong evidence (\(\chi^2 = 22.374\); two-tailed \(P < 0.001\)) that the odds in the population of having a pet bird is not the same for people with lung cancer (odds: \(0.695\)) and for people without lung cancer (odds: \(0.308\); OR: \(2.26\); \(95\)% CI from \(1.6\) to \(3.2\)).
Ex. 35.10. 1. RQ could be worded in terms of odds, odds ratios, proportions or associations. \(H_0\): \(\text{pop. odds, vegetarians} = \text{pop. odds non-vegetarians}\); \(H_1\): \(\text{pop. odds, vegetarians} \ne \text{pop. odds non-vegetarians}\). 2. The pop. OR, comparing the odds of being B12 deficient, for vegetarians to non-vegetarians. 3. \(\chi^2 = 4.707\). 4. Equivalent to \(z = 2.17\), so small-ish \(P\); \(P = 0.030\) from output. 5. The sample provides moderate evidence (\(\chi^2 = 4.707\); \(P = 0.030\)) that the odds in the population of being vitamin B12 deficient is different for vegetarian women (odds: \(0.3077\)) compared to non-vegetarian women (odds: \(0.0976\); OR: \(3.2\); \(95\)% CI: \(1.08\) to \(9.24\)). 6. Using the jamovi output, the smallest expected count is \(4.39\). Only one cell has an expected count less than \(5\), and only just; we shouldn't be too concerned (it should be noted).
Ex. 35.11. 1. \(H_0\): No association; \(H_1\) An association. 2. \(23.0522\); \(P = 0.00004\). 3. \(4\). 4. \(z = 2.772\). 5. Very strong evidence of an association. 6. Yes.
Ex. 35.12. 1. \(H_0\): No association; \(H_1\) An association. 2. \(0.800\), \(0.725\), \(0.710\). 3. \(4\), \(2.642\), \(2.444\). 4. \(1.637\), \(1.081\); reference level. 5. \(1.0989\); \(P = 0.577\). 6. \(2\). 7. \(0.741\). 8. No evidence of an association. 9. Yes.
Chap. 36: Selecting an analysis
Ex. 36.1. Summary of mean diffs.; histogram of diffs. Paired samples \(t\)-test; CI for mean diff.
Ex. 36.2. Summary of the two groups, plus diff. between means; boxplot, error-bar chart. CI for the diff. between two means.
Ex. 36.3. Comparing two odds: odds ratios; stacked, side-by-side bar chart. CI for odds ratio.
Ex. 36.4. A summary of two groups (over \(30\); \(30\) and under), plus the diff. between the means; boxplot and error-bar chart. A \(t\)-test and CI for the diff. between two means.
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. \(n > 25\).
Ex. 37.2. 1. \(H_0\): \(\rho = 0\); \(H_1\): \(\rho \ne 0\). 2. Moderate evidence of a relationship. 3. \(H_0\): \(\rho = 0\); \(H_1\): \(\rho \ne 0\). 4. No evidence of a relationship. 5. Linearity; \(n > 25\); constant variation in pesticide levels.
Ex. 37.3. The plot looks linear; \(n = 25\); variation not constant.
Ex. 37.4 The plot looks non-linear for larger gestational ages; \(n > 25\); 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 Phu Quoc ridgeback dogs also taller (\(r = 0.837\); one-tailed \(P < 0.001\); \(n = 30\))'. 4. Approx. linear; variation approx. constant; \(n > 25\): statistically valid.
Ex. 37.6. 1. Very close to \(-1\). 2. \(r = -\sqrt{0.9929} = -0.9964\). (\(r\) must be negative!) 3. Very small; a very large value for \(r\) on reasonable sized sample. 4. Yes.
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. 37.8. \(H_0\): \(\rho = 0\) and \(H_1\): \(\rho \ne 0\). \(P\)-value is \(0.190\): no evidence of a relationship.
Ex. 37.9. \(H_0\): \(\rho = 0\); \(H_1\): \(\rho \ne 0\). \(P < 0.001\): very strong evidence of a relationship.
Ex. 37.10. 1. Close to \(-1\), but not super close. 2. \(r = -0.819\). (\(r\) must be negative). 3. Very small; a large value for \(r\) on a reasonable sized sample. (The \(P\)-value actually is \(0.000104\).) 4. \(n < 25\); test may not be stat. valid.
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. 38.2. 1. Way too many decimal places. \(r\) not relevant: relationship non-linear. 2. Regression inappropriate: relationship non-linear. 3. \(y\) should be \(\hat{y}\); slope, intercept values swapped. 4. The whole thing is bothersome...
Ex. 38.3. 1. \(b_0 = 3.5\); \(b_1 = -0.14\). 2. \(b_0 = 2.1\); \(b_1 = -0.0047\).
Ex. 38.4. 1. \(b_0 = -1.03\); \(b_1 = 7.2\). 2. \(b_0 = -0.46\); \(b_1 = -1.88\).
Ex. 38.5. 1. \(r = 0.264\). 2. \(R^2 = 6.97%\); using neck circumference reduces the unknown variation by about \(7\)%. 3. \(\hat{y} = -24.47 + 1.36x\), where \(y\) is the REI and \(x\) the 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. 38.6. 1. \(\hat{y} = 150.19 - 0.348x\) (\(y\): mean number of ED patients; \(x\): number of days since welfare distribution). 2. Each extra day after welfare distribution associated with decrease in mean number of ED patients of about \(0.35\). Perhaps easier: Each \(10\) extra days after welfare distribution associated with decrease in mean number of ED patients of about \(10\times 0.35 = 3.5\). 3. \(-0.441\) to \(-0.255\) patients per day. 4. \(t = -7.45\); two-tailed \(P\)-value very small: \(P < 0.001\).
Ex. 38.7. 1. Intercept not about \(110\); that's where the line 'stops', but the intercept is the predicted value of \(y\) when \(x = 0\). We have to extend the line quite a bit. Using rise-over-run, guess slope is \((190 - 110)/(180 - 110) = 1.14\). 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 foot. 4. \(H_0\): \(\beta = 0\); \(H_1\): \(\beta \ne 0\). (You could answer in terms of correlations.) Question stated as two-tailed question, but testing if stronger legs increase kicking distance seems sensible. 5. \(t = 6.16\), which is 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. 38.8. No answer (yet).
Ex. 38.9. 1. \(\hat{y} = 17.47 - 2.59x\), where \(x\) is the percentage bitumen by weight, and \(y\) 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 is strong. 5. \(\hat{y} = 1.909\), or about \(1.9\)%. Might be a poor prediction, since this is extrapolation.
Ex. 38.10. No answer (yet).
Ex. 38.11. No answer (yet).
Ex. 38.12. 1. \(b_0\): No time spent on sunscreen application, average of \(0.27\) g has been applied; nonsense. \(b_1\): Each extra minute spent on application adds an average of \(2.21\) g of sunscreen: sensible. 2. \(\beta_0\) could be zero... which would make sense. 3. \(\hat{y} = 18\) g. 4. About \(64\)% of the variation in sunscreen amount applied can be explained by the variation in the time spent on application. 5. \(r = 0.8\) (must be positive value). A strong positive correlation between the variables.
Ex. 38.13. 1. Too many decimal places! Regression equation implies predicting \(0.0001\) of a gram. \(r\) has too many decimal places too. 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 increase in chest circumference by one cm. 5. Intercept: cm; slope: cm/gram. 6. \(\hat{y} = 2538.7\)g.
Chap. 39: Writing research
Ex. 39.1. 1. to. 2. its. 3. Only one sample with \(50\) individuals; use 'mean' or 'median', as appropriate, instead of 'average'. 4. Should be joined as one sentence.
Ex. 39.2. 1. their. 2. 'Sample' is singular: The sample was taken. 3. Effect. 4. Better than what? Better in what way (strength? durability?)
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. 39.4. 1. Ambiguous; sounds like field was liquid. 'Liquid fertiliser was applied to one of the fields.' 2. Diets do not lose weight; people lose weight; 'average' is vague. 'The mean weight loss was greater on the new diet compared to the traditional diet.'
Ex. 39.5. Number decimal places ridiculous.
Ex. 39.6. 'More likely' than what? Everybody will die.
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. 39.8. No units of measurement; jump-heights given to \(0.001\) of a centimetre.; table could also summarise information for each individual jump type; numerical summary shouldn't include \(P\)-value, \(t\)-score, or CI.
Ex. 39.9. Variables qualitative: means inappropriate; appropriate summary is odds ratio, so values almost certainly refer to the CI for the OR. Without more information, we can't really be sure what the OR means though.
Ex. 39.10. This study alone cannot prove anything; difference between hang times is of interest: appropriate CI is for difference between the mean hang times.
Chap. 40: Reading research
Ex. 40.1. 1. Convenience; self-selected. Nothing obvious suggests those in the study would 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: is 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. 40.2. 1. Students at that university (QUMS). 2. Observational. 3. A random sampling method has been used, so results should be generalisable to the population (students at that university). 4. Double-barrelled. 5. \(\hat{p} = 0.8603\); \(\text{s.e.}(\hat{p}) = 0;011781\); from \(0.837\) to \(0.884\). 6. \(t\)-test comparing two means. 7. Null: mean HQL score same for females and males. 8. \(\text{s.e.}(\text{diff}) = 0.2375\). 9. \(t = -2.61\); expect small \(P\); evidence mean different for females and males.10. \(0.145\) to \(1.085\), higher for males. 11. Yes. 12. Null: mean score the same in all three groups (mno difference'). Alternative: mean score is not the same in all three groups. 13. Only earphone users. 14. Evidence that mean not the same in all three groups. 15. \(19.6\) to \(20.0\). 16. \(0.20488\). 17. \(t = 3.90\); small \(P\), as in table.
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 a difference. 7. Probably, as the given standard errors are rounded. 8. Conceptual. 9. Table not shown. 10. \(\chi^2\). 11. \(z = 1.75\); \(P\) between \(5\)% and \(32\)%, which is not helpful. 12. Observational, so not cause-and-effect; no confounders noted; very restricted population.
Ex. 40.4. 1. \(r = 0.52\). 2. About \(27\)% of variation in Se concentration explained by electrical conductivity. 3. Very strong evidence that population slope is not zero. 4. \(\mu\)g.m.dS\(-1\).L\(-1\)^. 5. Many correct forms; proportion of wells with Se concentration less than \(2\) \(\mu\)g.L\(-1\) same for areas with and without Pliocene rocks within \(5\) km. 6. \(5.61\). 7. Very small. 8. Very strong evidence that the proportion of wells with Se concentration less than \(2\) \(\mu\)g.L\(-1\) different for areas with and without Pliocene rocks within \(5\) km.
Ex. 40.5. 1. \(\chi^2\)-test to compare proportions. 2. No evidence of a difference in survival rates at the two 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 that the mean sizes at the two temperatures were different; for Ae., no evidence that 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 a linear association. 13. Need scatterplot to be sure, but \(n > 25\). 14. When the predator--size ratio increase by one, predation efficiency increases by \(31.64\) percentage points. 15. About \(8.7\)% of the variation in predation efficient can be explained by the value of the predator--size ratio. 16. \(r = 0.294\).
Ex. 40.6. 1. Hardly! People may have tried to open their mouth far wider than usual. 2. Evidence of relationship between height and MMO; moderate positive correlation (taller associated with larger MMO, in general). 3. \(R^2 = 29\)%; about \(29\)% of variation in MOO explained by height. 4. Slope: Each extra \(10\) cm of height associated with \(3.6\) mm greater MMO, on average. 5. \(\hat{y} = 54.29\). 6. Two-sample \(t\)-test. 7. Extremely small. 8. Need to see graph; \(n > 25\). 9. Very strong evidence males have larger mean MMO than females. 10. Yes; relationship between height and MMO may be because males are taller on average. 11. Population is quite restricted. Neither individuals nor assessors blinded. Non-random sample. May be others.
Ex. 40.7. 1. Stratified? 2. Two-sample \(t\)-test. 3. Strong evidence that the mean number of actinomycetes are different. 4. Possibly not; sample sizes small (does not mean results are 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.
Ex. 40.8. 1. To act as a control, so any increase has a reference value. 2. Experimental. 3. Objective measures, to aid with managing placebo effect. 4. To ensure the two comparison groups included women of similar age (to help manage confounding). 5. \(0.92208\). 6. \(t = 0.542\). 7. Yes. 8. No evidence of a difference in mean age. 9. Yes. 10. To ensure the two comparison groups included women in similar rooms (to help manage confounding). 11. Not shown. 12. \(z - 0.592\). 13. Yes. 14. No evidence of a difference in percentage of room with air-con. 15. Yes; all expected counts will exceed \(5\). 16. Paired \(t = 8.93\); very small \(P\). 17. Very strong evidence of a mean increase. 18. Paired \(t = 5.33\); small \(P\). 19. Very strong evidence of a mean increase. 20. Hawthorne effect. 21. \(\text{s.e.}( \bar{x}_T - \bar{x}_T) = 4.377976\). 22. \(-2.16\) to \(15.36\) mins, greater for treatment group. 23. \(t = 1.51\); one-tailed \(P\) not very small; no difference between mean increase in sleep times. 24. Yes.