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Ask Good Questions: A blog about teaching introductory statistics HYPERLINK "" Post #25: Group quizzes, part 1 HYPERLINK "" #26: Group quizzes, part 2 HYPERLINK "" Feel free to use and/or modify the following quiz questions and solutions for use with your students.1. For parts (a) and (b), consider the research question of whether students at Cal Poly – San Luis Obispo are more likely to wear clothing that says “Cal Poly” than students at Cal Poly – Pomona. Suppose that you were to collect data for a statistical study of this question. a) Identify the explanatory variable, and classify it as numerical or categorical.b) Identify the response variable, and classify it as numerical or categorical.For parts (c) – (e), consider the patients who went to the emergency room at the local hospital last week as the observational units in a statistical study.c) Identify one categorical variable that could be recorded on these observational units.d) Identify one numerical variable that could be recorded on these observational units.e) State a research question that you could investigate about these observational units, using at least one of the variables that you gave in part (c) or (d).a) The explanatory variable is the Cal Poly campus attended by the student. This is a binary, categorical variable.b) The response variable is whether or not the student wears clothing that says “Cal Poly.” This is also a binary, categorical variable.c) Many answers are possible. Some examples include the person’s gender, whether or not the person arrived in an ambulance, and whether or not the person was admitted to the hospital.d) Many answers are possible. Some examples include the person’s age, how long the patient had to wait before seeing a doctor, and the cost of the treatment.e) Many answers are possible. Some examples include whether those who arrive in an ambulance tend to wait less long than those who do not arrive in an amblance, whether men or women or more likely to require stitches, and whether younger people have to wait longer than older people.2. Recall that you took samples of words from the population of words in the Gettysburg Address, for which the average length of a word is 4.295 letters. Parts (a)-(d) refer to this situation.a) Is the number 4.295 a parameter or a statistic? b) When you first selected your sample of 10 (circled) words, what was the variable? Was it categorical or numerical?c) What aspect of the first graph on the board indicated that the sampling method was biased?d) Would selecting words by closing your eyes and pointing at the page 10 times produce an unbiased sampling method? Explain briefly.e) In general, does taking a very large sample (say, of millions of people) produce an unbiased sampling method? Explain briefly.a) The number 4.295 is a parameter, because it describes the entire population of all words in the speech.b) The variable was word length (number of letters in the word), which is numerical.c) The bias was evident in that most of the sample means (a large majority) were greater than the population mean, indicating that the sampling method systematically favored larger words over smaller words.d) No. This sampling method would also systematically favor larger words over smaller words, because larger words take up more space on the page and so would be more likely to be selected than smaller words.e) No. It’s certainly possible to select a very large sample in a biased manner. The Literary Digest survey of 1936 is a good example of this, as are many online polls.3. Researchers investigated whether they could correctly predict the outcome of an election, more often than not, by selecting the candidate whose face is judged (by a majority of people interviewed) to be more competent-looking. They applied this prediction method to 32 U.S. Senate races in 2004. The “competent face” method correctly predicted the winner in 23 of the 32 races.a) What are the observational units in this study, and what is the sample size?b) Describe (in words) the null hypothesis to be tested.Consider the following results of a simulation analysis with 10,000 repetitions, for testing whether the competent face method would correctly predict the winner in more than half of all races:c) Describe how you would use the simulation results to approximate the p-value of the test.d) The p-value turns out to be approximately 0.01. Write a sentence interpreting this p-value in context (probability of what, assuming what?).e) Do the sample data provide strong evidence in support of the “competent face” prediction method? Justify your answer, based on the simulation analysis.a) The observational units are the 32 Senate races. The sample size is 32.b) The null hypothesis is that the “competent face” method is not useful for predicting election winners, so it would make correct predictions in 50% of all races in the long run.c) The p-value can be approximated by counting the number of repetitions that resulted in 23 or more successes, and then dividing by the number of repetitions (10,000).d) This p-value is the probability that the method would correctly predict 23 or more races correctly in a sample of 32 races, assuming that the method has a .5 probability of a correct prediction for every race.e) This analysis reveals that it would be quite surprising to correctly predict 23 or more of 32 election races if the “competent face” method were no better than flipping a coin. More specifically, such an extreme result would only occur about 1% of the time. So, the data provide fairly strong evidence in support of the researchers’ conjecture that the “competent face” method works more than half the time.4. Suppose that a tire manufacturer believes that the lifetimes of its tires follow a normal distribution with mean 50,000 miles and standard deviation 5,000 miles.a) Based on the empirical rule, 95% of tires last for between what two values?b) How many standard deviations above the mean is a tire that lasts for 58,500 miles?c) Determine the probability that a randomly selected tire lasts for more than 58,500 miles.d) Determine the mileage for which only 25% of all tires last longer than that mileage. Show how you arrive at your answer.e) Suppose the manufacturer wants to issue a money-back guarantee for its tires that fail to achieve a certain number of miles. If they want 99% of the tires to last for longer than the guaranteed number of miles, how many miles should they guarantee? Show how you arrive at your answer.a) The empirical rule says that 95% of values in a normal distribution fall within ±2 SDs of the mean, which is 50,000 ± 2(5000), which is 40,000 to 60,000 miles.b) The z-score is (58,500 – 50,000)/5000 = 1.7, so a tire that lasts for 58,500 miles is 1.7 standard deviations above the mean.c) The normal probability table reports a probability of .9554 for a z-score of 1.70, so the probability of lasting for more than 57,500 miles is 1 – .9554 = .0446.d) We need 25% of all tires to last for longer than this mileage, so we need 75% to last for less long. Looking in the normal probability table for an area of .7500, or as close as we can find, reveals that the corresponding z-score is 0.67, so the desired mileage is 0.67 SDs above the mean: 50,000 + 0.67(5000) ≈ 53,350 miles.e) We need 1% of all tires to last for less than this mileage, so we look in the table for an area of .0100, or as close as we can find. The corresponding z-score is -2.33, so the guaranteed mileage is 2.33 SDs below the mean: 50,000 – 2.33(5000) ≈ 38,350 miles.5. Recall the formula for the z-test statistic when conducting a hypothesis test about a proportion: z=p-π0π0(1-π0)n .a) What does the symbol p represent? (Be as specific as possible.)b) What does the symbol π0 represent? (Be as specific as possible.)c) What does the symbol n represent?d) For a given value of n, what happens to the absolute value of the test statistic as the difference between p and π0 increases?e) For a given value of n, what happens to the p-value as the difference between p and π0 increases?a) The symbol p represents the sample proportion having the characteristic of interest.b) The symbol π0 represents the hypothesized value of the population proportion having the characteristic of interest.c) The symbol n represents the sample size (the number of observational units in the sample).d) The absolute value of the test statistic gets larger.e) The p-value gets smaller.6. A Harris Poll that surveyed 2225 adult Americans on October 14-19, 2015 found that 29% reported having at least one tattoo.a) Is 29% (.29) a parameter or a statistic? What symbol do we use for it?b) Determine (by hand) a 95% confidence interval for the relevant parameter.c) Interpret this interval: You’re 95% confident that __________ is between ____ and ____ .d) How would a 99% confidence interval differ (if at all) from the 95% one? Comment on both the midpoint and width of the interval. (Do not bother to calculate a 99% confidence interval.)e) The same Harris Poll also found that 47% of respondents between the ages of 18-35 reported having at least one tattoo. How would a 95% confidence interval for this age group, based on this survey, compare to the 95% confidence interval that you found in part (b)? Comment on both the midpoint and width of the interval.a) This is a statistic: p = .29b) .29 ± 1.96.29(1-.29)2225, which is .29 ± 1.96(.00962), which is .29 ± .019, c) We are 95% confident that the population proportion of all American adults who have at least one tattoo is between .271 and .319.d) A 99% CI would be wider but have the same midpoint (.29).e) The sample size would be smaller, because 18-35-year-olds comprise a subset of the original sample of 2225 adults. Therefore, the CI would be wider, and the midpoint would be larger (.47 rather than .29).7. Answer the following:a) Suppose that a class of 10 students has the following exam scores: 60, 70, 50, 60, 90, 90, 80, 80, 40, 50. Determine the median of these 10 exam scores.b) Suppose that the average amount of sleep obtained by Cal Poly undergraduate students last night was 6.8 hours, and the average amount of sleep obtained by Cal Poly graduate students last night was 7.6 hours. Is it reasonable to conclude that the average amount of sleep obtained last night among all Cal Poly students was (6.8 + 7.6)/2 = 7.2 hours? Explain.c) What effect does doubling every value in a dataset have on the mean? Explain your answer.d) What effect does adding 5 to every value in a dataset have on the standard deviation? Explain your answer.e) Create an example of 10 hypothetical exam scores (on a 0 – 100 scale) with the property that the mean is at least 20 points larger than the median. Also report the values of the mean and median for your example.a) The sorted quiz values are: 40, 50, 50, 60, 60, 70, 80, 80, 90, 90. The median is the average of the 5th and 6th ordered values, which is (60 + 70) / 2 = 65.b) No, this is not reasonable. This would only be true if the number of undergraduates is the same as the number of grad students. Cal Poly has many more undergraduates than grad students, so the overall average would be much closer to 6.8 hours than 7.2 hours.c) The mean would be doubled, because the sum of the values would be doubled.d) The standard deviation would not change. The mean would increase by 5, so the deviation between each value and the mean would remain the same for each value.e) Many examples are possible. The key is to have a few values that are much larger than the majority of values. An extreme example that satisfies the condition is: 0, 0, 0, 0, 0, 0, 100, 100, 100, 100. The mean score for this example is 40, and the median is 0. 8. Suppose that the mean age (in years) of all pennies currently in circulation in the U.S. is 12.3 years, and the standard deviation of these ages is 9.6 years. Suppose also that you take a random sample of 50 pennies and calculate the mean age of the pennies in your sample.a) Are the numbers 12.3 and 9.6 parameters or statistics? Explain briefly.b) Describe the sampling distribution of the sample mean penny age. Also produce a well-labeled sketch of this sampling distribution. c) Determine the probability that the sample mean age of your random sample of 50 pennies would be less than 10 years. (Show your work.)d) Are your answers to parts (b) and (c) approximately valid even if the distribution of penny ages is not normally distributed? Explain.e) Based on the values of the mean and standard deviation of penny ages, there is reason to believe that the distribution of penny ages is not normally distributed. Explain why.a) These are parameters, because they refer to the entire population of all pennies in circulation.b) Because the sample size (n = 50) is sufficiently large, the CLT establishes that the sampling distribution of the sample mean penny age is approximately normal with mean 12.3 years and SD 9.6/50 ≈ 1.358 years, as shown in this sketch:c) The z-score is (10 – 12.3) / 1.358 ≈ –1.69. The area to the left of this z-score under the standard normal curve is .0455.d) Yes, because the sample size (n = 50) is sufficiently large for the CLT to hold.e) The SD (9.6) is almost as large as the mean (12.3). In fact, the age 0 years has a z-score of (0 – 12.3) / 9.6 ≈ –1.28. Obviously no pennies can have a negative age, yet the standard normal curve has a good bit of area (about 10%) to the left of a z-score of –1.28.9. A study conducted in Dunedin, New Zealand investigated whether wearing socks over shoes could help people to walk confidently down an icy footpath*. Volunteers were randomly assigned either to wear socks over their usual footwear or to simply wear their usual footwear, as they walked down an icy footpath. An observer recorded whether or not the participant appeared to be walking confidently.a) Is this an observational study or an experiment? Explain briefly.b) Identify the explanatory and response variables.c) Does this study make use of random sampling, random assignment, both, or neither?d) Did the researchers use randomness in order to give all walkers in New Zealand the same chance of being selected for the study? Answer YES or NO.e) Did the researchers use randomness in order to produce groups that were as similar as possible in all respects before the explanatory variable was imposed? Answer YES or NO.a) This is an experiment, because the researchers (randomly) assigned participants to wear socks over their shoes or not.b) The explanatory variable is whether or not the walker wore socks over his/her shoes or not. The response variable is whether or not the walker appeared to be walking confidently.c) This study makes use of random assignment but not random sampling.d) No, the subjects in the study were volunteers, not randomly selected.e) Yes, the researchers randomly assigned subjects to groups in an effort to produce similar groups prior to imposing socks over shoes (or usual footwear).10. Recall that a study conducted in Dunedin, New Zealand investigated whether wearing socks over shoes could help people to walk confidently down an icy footpath*. Participants were randomly assigned to wear socks over their usual footwear, or to simply wear their usual footwear, as they walked down an icy footpath. One of the response variables measured was whether an observer considered the participant to be walked confidently. Results are summarized in the table below:Socks over shoesUsual footwearTotalAppeared to be confident10818Did not appear to be confident4711Total141529For parts (a) – (c), suppose that you conduct a by-hand simulation analysis to investigate whether wearing socks over shoes increases people’s confidence while walking down an icy footpath. For parts (d) and (e), consider the results of such a simulation analysis performed with technology.a) What would be the assumption involved with producing the simulation analysis? Choose one of the following options: A. That wearing socks over shoes has no effect on walkers’ confidence; B. That wearing socks over shoes has some effect on walkers’ confidence; C. That walkers are equally likely to feel confident or not, regardless of whether they wear socks over shoes or not; D. That walkers are more likely to feel confident if they wear socks over shoesb) How many cards would you use in the simulation analysis? What would the color breakdown be? c) How many cards would you deal out into groups? How many times would you repeat this process?d) The graph below displays the results of a simulation analysis with 10,000 repetitions, displaying the distribution of the difference in success proportions between the two groups. Describe how you would calculate an approximate p-value from this graph (i.e., where would you count?).e) Based on the 2×2 table of data and on this graph of simulation results, how much evidence do the data provide in support of the conjecture that wearing socks over shoes increases people’s confidence while walking down an icy footpath? Choose one of the following options: A. little or no evidence; B. moderate evidence; C. strong evidence; D. very strong evidence.a) A. That wearing socks over shoes has no effect on walkers’ confidence b) You would need 29 cards, 18 of one color (representing those who walked confidently) and 11 of another color (representing those who did not walk confidently).c) You would deal out 14 cards into one group (representing those who walked with socks over shoes) and 15 cards into another group (representing those who walked with usual footwear). This would be repeated for a very large number (say, 1000 or 10,000) repetitions.d) The observed value of the statistic (difference in success proportions between the two groups) is 10/14 – 8/15 ≈ .714 - .533 ≈ .181. The p-value is the number of repetitions with a difference in success proportions of .181 or greater, divided by the total number of repetitions (10,000). e) A. Little or no evidence. The p-value looks to be approximately .25. The observed outcome (10 successes in the “socks over shoes” group is not surprising if the null (that wearing socks over shoes has no effect on walking confidence) were true.11. Researchers at Stanford University studied whether a curriculum could help to reduce children’s television viewing. Third and fourth grade students at two public elementary schools in San Jose were the subjects. One of the schools, chosen at random, incorporated an 18-lesson, 6-month classroom curriculum designed to reduce watching television and playing video games, whereas the other school made no changes to its curriculum. At the beginning and end of the study, all children were asked to report how many hours per week they spent on these activities, both before the curriculum intervention and afterward. The tables below summarize reported amounts of television watching, first at the beginning of the study and then at its conclusion:Beginning of research studySample sizeSample meanSample SDControl group10315.4615.02Intervention group9515.3513.17Conclusion of research studySample sizeSample meanSample SDControl group10314.4613.82Intervention group958.8010.41a) Is the response variable in this study categorical or numerical?b) The difference between the groups can be shown not to be statistically significant at the beginning of the study. Do you think the researchers would be pleased by this result? Explain why or why not.c) Even if the distributions of reported amounts of television watching per week are sharply skewed, would it still be valid to apply a two-sample t-test on these data? Explain briefly.d) Calculate the value of the test statistic for investigating whether the two groups differ with regard to average amount of television watching per week.e) Based on the value of the test statistic, summarize your conclusion for the researchers.a) The response variable is the number of hours of television watching per week, which is quantitative.b) The researchers would hope that the two groups are as similar as possible before the treatments are imposed, so they should be pleased by this result (lack of a statistically significant difference between the groups).c) Yes, because the sample sizes (103 and 95) are very large.d) The test statistic is: t = xcontrol-xinterventionscontrol2ncontrol+sintervention2nintervention = 14.46-8.8013.822103+10.41295 ≈ 3.27e) The test statistic (t = 3.27) is large enough that the p-value is less than .005. The sample data provide very strong evidence that children in the intervention group tend to watch less television, on average, compared to children in the control group.12. Answer the following:a) Would you expect to find a positive or negative correlation coefficient between high temperature on January 1, 2020 and distance from the equator, for a sample consisting of the largest city in each of the 50 U.S. states? Explain briefly.b) Suppose that you record the daily high temperature and the daily amount of ice cream sold by an ice cream vendor at your favorite beach this summer, starting on the Friday of Memorial Day weekend and ending on the Monday of Labor Day weekend. Would you expect to find a positive or negative correlation coefficient between these variables? Explain briefly.c) Suppose that every student in this class scored 5 points lower on the second exam than on the first exam. Consider the correlation coefficient between first exam score and second exam score. What would the value of this correlation coefficient be? Explain briefly.Parts (d) and (e) pertain to the graph below, which displays data on the age (in months) at which a child first speaks and the child’s score on an aptitude test taken later in childhood:d) Is the value of the correlation coefficient between these variables positive or negative?e) Suppose that the child who took 42 months to speak were removed from the analysis. Would the value of the correlation coefficient between the variables be closest to -1, 0, or 1?a) Negative, because cities with a larger distance from the equator are farther north and therefore more likely to have a smaller temperature than cities with a smaller distance from the equator.b) The correlation coefficient would most likely be positive, because people probably buy more ice cream when the temperature is higher and less ice cream when the temperature is lower.c) The correlation coefficient would be 1.0, because the points would fall exactly on a straight line with a positive slope.d) Negativee) Close to 0. Without the outlier who took 42 months to speak, the association between the variables is still (slightly) negative but very weaker.13. Some of the statistical inference procedures that we have studied include:A.One-sample z-procedures for a proportionB.Two-sample z-procedures for comparing proportionsC. One-sample t-procedures for a meanD.Two-sample t-procedures for comparing meansE.Paired-sample t-procedures for comparing meansFor each of the following questions, identify (by capital letter) which procedure you would use to investigate that question. (Be aware that some letters may be used more than once, others not at all.)a) Do cows tend to produce more milk if their handler speaks to them by name every day than if the handler does not speak to them by name? A farmer randomly assigned half of her cows to each group and then compared how much milk they produced after one month.b) A baseball coach wants to investigate whether players run more quickly from second base to home plate if they take a wide angle or a narrow angle around third base. He recruits 20 players to serve as subjects for a study. Each of the 20 players runs with each method (wide angle, narrow angle) once.c) Does the average length of a sentence in a novel written by John Grisham exceed the average length of a sentence in a novel written by Louise Penny? Students took a random sample of 100 sentences from each author’s most recent novel and recorded the number of words in each sentence.d) Have more than 25% of Cal Poly students have been outside of California in the year 2019?e) Are Stanford students more likely to have been outside of California in the year 2019 than Cal Poly students? a) D (The explanatory variable is whether or not the cows were spoken to by name, which is categorical and binary. The response variable is amount of milk produced, which is numerical. Random assignment establishes that the data are not paired.)b) E ((The explanatory variable is whether the player takes a narrow or wide angle, which is categorical and binary. The response variable is time to run from second base to home plate, which is numerical. Because each player runs both methods, the data are paired.)c) D (The explanatory variable is which author wrote the sentence, which is categorical and binary. The response variable is sentence length, which is numerical. Random sampling was conducted independently, so the data are not paired.)d) A (The variable is whether or not the student has been outside of California in 2019, which is categorical and binary.)e) B (The explanatory variable is whether the student attends Cal Poly or Stanford, which is categorical and binary. The response variable is whether or not the student has been outside of California in 2019, which is also categorical and binary.) ................
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