Name: ________________________ Class: ___________________ Date: __________
ID: A
Chapter 10 review Multiple Choice Identify the choice that best completes the statement or answers the question. ____
1. According to Humane Society data, 39% of households in the United States have at least one dog. In the United Kingdom, 23% of households have at least one dog. Suppose you select an SRS of 75 households in the U.S. and 80 households in the U.K., and calculate = the proportion of households in the U.S. sample that have a dog, and
= the proportion of households in the U.K. sample that have a dog. Which of the
following best describes the sampling distribution of proportions for ? A. Mean = 0.16, Standard deviation = 0.024, Shape unknown B. Mean unknown, Standard deviation unknown, Shape approximately Normal. C. Mean = 0.16, Standard deviation = 0.024, Shape unknown D. Mean unknown, Standard deviation unknown, Shape approximately Normal. E. Mean = 0.16, Standard deviation = 0.073, Shape approximately Normal. ____
2. You have two large bins of marbles. In bin A, 40% of the marbles are red. In bin B, 52% of the marbles are red. You select a simple random sample of 30 marbles from bin A and 40 marbles from bin B. What is the probability that the proportion of red marbles in the sample from bin A is greater than the proportion of red marbles from bin B? A. nearly zero B. 0.0010 C. 0.1190 D. 0.1357 E. 0.1562
1
Name: ________________________ ____
ID: A
3. According to recent polls, 24% of people in the United States answered Yes to the question, “Did you smoke any form of tobacco yesterday?” In the European Union, 28% of people answered yes to a similar question. Let’s assume these are population parameters for the two populations. If you select a simple random sample of 40 people in the U.S. and 50 people in the E.U., which of the following expressions represents the standard deviation of the sampling distribution for the difference in the proportion of smokers in the two groups? A.
B.
C.
D.
E. ____
4. A consumer group has determined that the distribution of life spans for gas ranges (stoves) has a mean of 15.0 years and a standard deviation of 4.2 years. The distribution of life spans for electric ranges has a mean of 13.4 years and a standard deviation of 3.7 years. Both distributions are moderately skewed to the right. Suppose we take a simple random sample of 35 gas ranges and a second SRS of 40 electric ranges. Which of the following best describes the sampling distribution of , the difference in mean life span of gas ranges and electric ranges? A. Mean = 1.6 years, standard deviation = 7.9 years, shape: moderately right-skewed. B. Mean = 1.6 years, standard deviation = 0.92 years, shape: approximately Normal. C. Mean = 1.6 years, standard deviation = 0.92 years, shape: moderately right skewed. D. Mean = 1.6 years, standard deviation = 0.40 years, shape: approximately Normal. E. Mean = 1.6 years, standard deviation = 0.40 years, shape: moderately right skewed.
____
5. At a large state university, the heights of male students who are interscholastic athletes is approximately Normally distributed with a mean of 74.3 inches and a standard deviation of 3.5 inches. The heights of male students who don’t play interscholastic sports (we’ll call them “non-interscholastics”) is approximately Normally distributed with a mean of 70.3 inches and a standard deviation of 3.2 inches. You select an SRS of 10 interscholastic athletes and 12 non-interscholastics. What is the probability that the sample mean of non-interscholastics is greater than the sample mean of interscholastic athletes? A. nearly 0 B. 0.0027 C. 0.0035 D. 0.9965 E. 0.9973 2
Name: ________________________ ____
ID: A
6. An SRS of 100 is taken from a Normal distribution with a mean of 25 and a standard deviation of 4, and an SRS of 85 is taken from a different Normal distribution with a mean of 40 and a standard deviation of 7. Which of the following expressions represents the standard deviation of the sampling distribution of the difference of means from these two samples? A.
B.
C.
D.
E.
3
Name: ________________________
ID: A
Scenario 10-1 In a large Midwestern university (with the class of entering freshmen being on the order of 6000 or more students), an SRS of 100 entering freshmen in 1993 found that 20 finished in the bottom third of their high school class. Admission standards at the university were tightened in 1995. In 1997, an SRS of 100 entering freshmen found that 10 finished in the bottom third of their high school class. Let p1 be the proportion of all entering freshmen in 1993 who graduated in the bottom third of their high school class, and let p2 be the proportion of all entering freshmen in 1997 who graduated in the bottom third of their high school class. ____
7. Use Scenario 10-1. Which of the following represents 99% confidence interval for p1 – p2? A.
B.
C.
D.
E. ____
8. Use Scenario 10-1. Is there evidence that the proportion of freshmen who graduated in the bottom third of their high school class in 1997 has been reduced as a result of the tougher admission standards adopted in 1995, compared to the proportion in 1993? To determine this, you test the hypotheses H0: p1 = p2, Ha: p1 > p2 at the α = 0.05 level. You calculate a test statistic of 1.980. Which of the following is the appropriate P-value and conclusion for your test? A. P-value = 0.047; fail to reject H0; we do not have evidence that the proportion who graduated in the bottom third of their class has been reduced. B. P-value = 0.047; accept Ha; there is evidence that the proportion who graduated in the bottom third of their class has been reduced. C. P-value = 0.024; fail to reject H0; we do not have evidence that the proportion who graduated in the bottom third of their class has been reduced. D. P-value = 0.024; reject H0; we have evidence that the proportion who graduated in the bottom third of their class has been reduced. E. P-value = 0.024; fail to reject H0; we have evidence that the proportion who graduated in the bottom third of their class has not changed.
4
Name: ________________________ ____
ID: A
9. Use Scenario 10-1. Which of the following best explains why it was important to know that the university had 6000 or more entering freshman before using z-procedures in this situation? A. If the size of the freshman classes were much smaller, we could not be confident that the Normality condition for these procedures had been met. B. The central limit theorem would not apply if the population size was below 500. C. To meet the independence condition for this procedure, we needed to know that the samples were less than 10% of the population size. D. To meet the random condition for this procedure, we needed to know that the samples were less than 10% of the population size. E. The information about the size of the Freshman classes was not important, it was added to the problem simply to provide extraneous numbers. Scenario 10-2 An SRS of 100 flights by Nite-flite Airlines showed that 64 were on time. An SRS of 100 flights by Waxwing Airlines showed that 80 were on time. Let pN be the proportion of on-time flights for all Nite-flite Airline flights, and let pW be the proportion of all on-time flights for all Waxwing Airlines flights.
____ 10. Use Scenario 10-2. A 95% confidence interval for the difference pA – pW is A. B. C. D. E. ____ 11. Use Scenario 10-2. Is there evidence of a difference in the on-time rate for the two airlines? To determine this, you test the hypotheses H0: p1 = p2, Ha: p1 p2. The P-value of your test is 0.0117. Which of the following is an appropriate interpretation of the P-value? A. If the on-time rates for the two airlines are equal, the probability of getting samples with a difference as far or farther from zero as these samples is 0.0117. B. If the on-time rates for the two airlines are not equal, the probability of getting samples with a difference as far or farther from zero as these samples is 0.9883. C. The probability of making a Type I error is 0.0117. D. The probability of making a Type II error is 0.0117. E. The probability that H0 is true is 0.0117.
5
Name: ________________________
ID: A
Scenario 10-3 A manufacturer receives parts independently from two suppliers. An SRS of 400 parts from supplier 1 finds 20 that are defective. An SRS of 100 parts from supplier 2 finds 10 that are defective. Let p1 and p2 be the proportions of all parts from suppliers 1 and 2, respectively, that are defective. ____ 12. Use Scenario 10-3. Which of the following represents a 95% confidence interval for p1 – p2? A.
B.
C.
D.
E.
____ 13. Use Scenario 10-3. Suppose the 95% confidence interval for the difference in the proportion of defective parts from the two suppliers is . Which of the following is the best interpretation of the confidence interval? A. 95% of the time, the true difference in the proportion of defective parts from the two suppliers is in the interval . B. We are 95% confident that the true difference in the proportion of defective parts from the two suppliers is in the interval . C. The probability that the true difference in the proportion of defective parts from the two suppliers is in the interval is 95%. D. We are 95% confident that the interval E.
captures the true difference in the
proportion of defective parts from the two suppliers. 95% of the differences calculated from samples this size will be in the interval
6
.
Name: ________________________
ID: A
Scenario 10-4 An agricultural researcher wishes to see if a kelp extract helps prevent frost damage on tomato plants. One hundred tomato plants in individual containers are randomly assigned to two different groups. Plants in both groups are treated identically, except that the plants in group 1 are sprayed weekly with a kelp extract, while the plants in group 2 are not. After the first frost in the autumn, 12 of the 50 plants in group 1 exhibited damage, and 18 of the 50 plants in group 2 showed damage. Let p1 be the actual proportion of all tomato plants of this variety that would experience damage under the kelp treatment, and let p2 be the actual proportion of all tomato plants of this variety that would experience damage under the no-kelp treatment, assuming that the tomatoes are grown under conditions similar to those in the experiment. ____ 14. Use Scenario 10-4. A 99% confidence interval for p1 – p2 is A. B. C. D. E. ____ 15. Use Scenario 10-4. Is there evidence of a decrease in the proportion of tomatoes suffering frost damage for tomatoes sprayed with kelp extract? To determine this, you test the hypotheses H0: p1 = p2, Ha: p1 < p2. The P-value of your test is A. greater than 0.10. B. between 0.05 and 0.10. C. between 0.01 and 0.05. D. between 0.001 and 0.01. E. below 0.001. ____ 16. Use Scenario 10-4. In the original design for this experiment, 50 tomato plants were grown in one large container and 50 more were grown in a second container. The plants in container 1 were sprayed with kelp extract and the plants in container 2 were not. Why would z-procedures for confidence intervals and tests of significance be of questionable value in this situation? A. We cannot be sure that the Normality condition has been met. B. We don’t know whether the 10% condition has been met. C. Individual plants were not assigned randomly to the two experimental treatments—any influence of kelp extract is confounded with differences between the two containers. D. Because we don’t know the standard deviation for either population, we shouldn’t use z-procedures. E. We are collecting results on the entire population of plants, so statistical inference from samples is unnecessary.
7
Name: ________________________
ID: A
Scenario 10-5 An SRS of 45 male employees at a large company found that 36 felt that the company was supportive of female and minority employees. An independent SRS of 40 female employees found that 24 felt that the company was supportive of female and minority employees. Let p1 represent the proportion of all male employees members at the company and p2 represent the proportion of all female employees members at the vs. company who hold this opinion. We wish to test the hypotheses ____ 17. Use Scenario 10-5. Which of the following is the correct expression for the test statistic? A.
B.
C.
D.
E.
8
Name: ________________________
ID: A
____ 18. Use Scenario 10-5. The P-value for this test is 0.0217. Which of the following is a correct conclusion? A. Reject H0 at α = 0.01: we have evidence that the proportion of male employees who feel that the company is supportive of women and minority employees is higher than the proportion of women who feel this way. B. Reject H0 at α = 0.01: we do not have evidence that the proportion of male employees who feel that the company is supportive of women and minority employees is higher than the proportion of women who feel this way. C. Accept H0 at α = 0.01: we do not have evidence that the proportion of male employees who feel that the company is supportive of women and minority employees is higher than the proportion of women who feel this way. D. Accept Ha at α = 0.01: we do not have evidence that the proportion of male employees who feel that the company is supportive of women and minority employees is higher than the proportion of women who feel this way. E. Fail to reject H0 at α = 0.01: we do not have evidence that the proportion of male employees who feel that the company is supportive of women and minority employees is higher than the proportion of women who feel this way.
9
Name: ________________________
ID: A
Scenario 10-6
Divorced within five years?
Child within three years? YES NO 83 52 137 128 220 180
YES NO Total
A sociologist hypothesizes that couples who have a child within the first three years of marriage are more likely to divorce. From city records, she selects a random sample of 400 couples who were both between the ages of 20 and 25 when they married. She compared the divorce rate of couples who had a child within the first three years of marriage to the divorce rate of couples who did not. Here are her results: ____ 19. Use Scenario 10-6. Let p1 = proportion of couples that had a child within the first three years and were divorced within five years and p2 = proportion of couples that did not have a child within the first three years and were divorced within five years. We wish to test the hypotheses vs. . Which of the following is the appropriate expression for the test statistic? A.
B.
C.
D.
E.
10
Name: ________________________
ID: A
____ 20. Use Scenario 10-6. The P-value for this one-sided test is 0.0314. If α = 0.05, which of the following is the best conclusion? A. there is evidence of an association between divorce rate and having children early in a marriage. B. having more children increases the risk of divorce during the first 5 years of marriage. C. If you want to decrease your chances of getting divorced, it is best to marry later in life. D. If you want to decrease your chances of getting divorced, it is best not to have children. E. If you want to decrease your chances of getting divorced, it is best to wait several years before having children. Scenario 10-7 Some researchers have conjectured that stem-pitting disease in peach tree seedlings might be controlled with weed and soil treatment. An experiment was conducted to compare peach tree seedling growth with soil and weeds treated with one of two herbicides. In a field containing 20 seedlings, 10 were randomly selected from throughout the field and assigned to receive Herbicide A. The remaining 10 seedlings were to receive Herbicide B. Soil and weeds for each seedling were treated with the appropriate herbicide, and at the end of the study period, the height (in centimeters) was recorded for each seedling. A box plot of each data set showed no indication of non-Normality. The following results were obtained:
(cm) Herbicide A Herbicide B
94.5 109.1
S (cm) 10 9
____ 21. Use Scenario 10-7. A 95% confidence interval for (Use the conservative value for degrees of freedom.) A.
B.
C.
D.
E.
11
is given by which of the following expressions?
Name: ________________________
ID: A
____ 22. Use Scenario 10-7. Suppose we wished to determine if there tended to be a significant difference in mean height for the seedlings treated with the different herbicides. To answer this question, we decide to test the vs. Ha: . Based on our data, which of the following is the value of hypotheses H0: test statistic? A. 14.60 B. 7.80 C. 3.43 D. 2.54 E. 1.14 is 14.6 ± 7.80. We wish to test the hypotheses
____ 23. The 90% confidence interval for the difference H0: A. B. C. D. E.
vs. Ha: . Based on this confidence interval we would not reject H0 at the 0.10 level. we would reject H0 at the 0.10 level. we would not reject H0 at the 0.05 level. we would reject H0 at the 0.05 level. we would accept Ha at the 0.10 level.
12
Name: ________________________
ID: A
Scenario 10-8 A researcher wished to test the effect of the addition of extra calcium to yogurt on the “tastiness” of yogurt. Sixty-two adult volunteers were randomly divided into two groups of 31 subjects each. Group 1 tasted yogurt containing the extra calcium. Group 2 tasted yogurt from the same batch as group 1 but without the added calcium. Both groups rated the flavor on a scale of 1 to 10, with 1 being “very unpleasant” and 10 being with a standard deviation of The mean “very pleasant.” The mean rating for group 1 was rating for group 2 was with a standard deviation of . Let and represent the mean ratings we would observe for the entire population represented by the volunteers if all members of this population tasted, respectively, the yogurt with and without the added calcium. ____ 24. Use Scenario 10-8. Assuming the conditions for using t-procedures have been met, which of the following represents a 90% confidence interval for (using a conservative value for the degrees of freedom). A.
B.
C.
D.
E.
____ 25. Use Scenario 10-8. Which of the following would lead us to believe that the t-procedures were not safe to use in this situation? A. The sample medians and means for the two groups were slightly different. B. The distributions of the data for the two groups were both slightly skewed right. C. The data are integers between 1 and 10 and so cannot be normal. D. The standard deviations from both samples were very different from each other. E. None of the above.
13
Name: ________________________
ID: A
____ 26. Use Scenario 10-8. If we had used the more accurate software approximation for the degrees of freedom, how would the 90% confidence interval compare to the one we constructed with the more conservative value for degrees of freedom? A. It would be wider. B. It would be narrower. C. It would not change. D. Whether it would be wider, narrower, or stay the same depends on the sample sizes. E. Since t confidence intervals are constructed with sample standard deviations, we don’t know whether it would be wider, narrower, or the same. Scenario 10-9 A sportswriter wished to see if a football filled with helium travels farther, on average, than a football filled with air. To test this, the writer used 18 adult male volunteers. These volunteers were randomly divided into two groups of nine subjects each. Group 1 kicked a football filled with helium to the recommended pressure. Group 2 kicked a football filled with air to the recommended pressure. The mean yardage for group 1 was yards with a standard deviation of yards. The mean yardage for group 2 was yards with a standard deviation of yards. Assume the two groups of kicks are independent. Let and represent the mean yardage we would observe for the entire population represented by the volunteers if all members of this population kicked, respectively, a helium- and an air-filled football. Dot plots of the two data sets show no indication of non-Normality. ____ 27. Use Scenario 10-9. Which of the following is a 99% confidence interval for µ1 – µ2 (using the conservative value for the degrees of freedom)? A. B. C. D. E. ____ 28. Use Scenario 10-9. The sportswriter wishes to test the hypotheses vs. . The P-value for the test (using the conservative value for the degrees of freedom) is 0.132. Which of the following is the appropriate conclusion to draw from this test, if µ = 0.05? A. Accept Ha B. Reject Ha C. Reject H0 D. Fail to reject Ha E. Fail to reject H0
14
Name: ________________________
ID: A
Scenario 10-10 A researcher wishes to compare the effect of two stepping heights (low and high) on heart rate in a step-aerobics workout. He randomly assigns 50 adult volunteers to two groups of 25 subjects each. Group 1 does a standard step-aerobics workout at the low height. The mean heart rate at the end of the workout for the subjects in group 1 was beats per minute with a standard deviation of beats per minute. Group 2 did the same workout but at the high step height. The mean heart rate at the end of the workout for beats per minute with a standard deviation of beats per the subjects in group 2 was minute. Assume the two groups are independent and both data sets are approximately Normal. Let µ1 and µ2 represent the mean heart rates we would observe for the entire population represented by the volunteers if all members of this population did the workout using the low or high step height, respectively. ____ 29. Use Scenario 10-10. Which of the following is a 98% confidence interval for value for the degrees of freedom)? A.
(using the conservative
B. C. D. E. ____ 30. Use Scenario 10-10. Which of the following is a correct interpretation of this interval? A. 98% of the time, the true difference in the mean heart rate of subjects in the high-step vs. low-step groups will be in this interval. B. We are 98% confident that this interval captures the true difference in mean heart rate of subjects in the high-step vs. low-step groups. C. There is a 0.98 probability that the true difference in mean heart rate of subjects in the high-step vs. low-step groups in this interval. D. 98% of the intervals construction this way will contain the value 0. E. There is a 98% probability that we have not made a Type I error. ____ 31. Use Scenario 10-10. The researcher decides to test the hypotheses
vs.
at
the α = 0.05 level and produces a P-value of 0.0475. Which of the following is a correct interpretation of this result? A. The probability that the difference is 0.0475. B. The probability that this test resulted in a Type II error is 0.0475. C. If this test were repeated many times, we would make a Type I error 4.75% of the time. D. If the null hypothesis is true, the probability of getting a difference in sample means as far or farther from 0 as the difference in our samples is 0.0475. E. If the null hypothesis is false, the probability of getting a difference in sample means as far or farther from 0 as the difference in our samples is 0.0475.
15
Name: ________________________
ID: A
____ 32. Use Scenario 10-10. The P-value in the previous question was produced by a calculator, using the software or estimate of 43.97 degrees of freedom. If we used the more conservative value of the smaller of , how would the P-value change for the same data? A. It would be smaller. B. It would be larger. C. It would not change, since the test statistic’s value is not influenced by the degrees of freedom. D. Since the P-value depends on the value of a random variable (the sample mean), we can’t predict whether it will be larger, smaller, or the same. E. Whether it’s larger, smaller, or the same depends on what level of significance we choose. ____ 33. An experiment to test the effectiveness of regular treatments with fluoride varnish to reduce tooth decay involved 36 volunteers who had half of their teeth—the right side or left side, determined by a coin flip—painted with a fluoride varnish every six month for 5 years. At the end of the treatments, the number of new cavities during the treatment period was compared on treatment (fluoride varnish) side versus the control (no fluoride varnish) side. The appropriate statistical test for analyzing the results of this experiment is A. One-sample z-test of proportions. B. Two-sample z-test for difference of proportions. C. One-sample t-test on paired data. D. Two-sample t-test for difference of means. E. Two-sample z-test for difference of means.
16
Name: ________________________
ID: A
____ 34. Does listening to music increase the speed at which people complete routine tasks? Fifteen volunteers are asked to sort 100 red and white beads into two piles according to color, once while listening to Handel’s Water Music and once in silence (the order—music or silence first—is determined for each subject by the flip of a coin). Here are the data (times are in seconds), along with summary statistics in the last two columns: Subject Handel Silence Difference
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
14
15
17
21
13
18
21
24
15
25
22
30
13
15
16
18.6
16
18
17
20
15
17
22
27
14
22
20
32
15
13
17
19.0
5.13
2
3
0
–1
2
–1
1
3
–1
–3
–2
2
2
–2
1
0.4
1.96
s 5.04
We wish to test the hypothesis that the mean difference in time to sort the beads with and without music is 0. Which of the following is the appropriate test statistic?
A.
B.
C.
D.
E.
17
ID: A
Chapter 10 review Answer Section MULTIPLE CHOICE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
ANS: ANS: ANS: ANS: ANS: TOP: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: TOP: ANS: ANS: ANS: ANS: ANS: ANS: TOP: ANS: ANS:
E PTS: 1 TOP: Sampling distribution of difference of proportions E PTS: 1 TOP: Probability calculation for sampling distribution of p1-p2 C PTS: 1 TOP: Sampling distribution of difference of proportions B PTS: 1 TOP: Sampling distribution of difference of means B PTS: 1 Probability calculation for sampling distribution of mu1-mu2 A PTS: 1 TOP: Sampling distribution of difference of means D PTS: 1 TOP: Confidence interval for p1-p2 (formula) D PTS: 1 TOP: Significance test for p1-p2: P-value and conclusion C PTS: 1 TOP: Conditions for inference p1-p2 E PTS: 1 TOP: Confidence interval for p1-p2 (computation) A PTS: 1 TOP: Significance test for p1-p2: Interpret P-value B PTS: 1 TOP: Confidence interval for p1-p2 (formula) D PTS: 1 TOP: Interpret confidence interval D PTS: 1 TOP: Confidence interval for p1-p2 (computation) B PTS: 1 TOP: P-value for test of significance C PTS: 1 TOP: Conditions for inference p1-p2 B PTS: 1 TOP: Significance test for p1-p2: test statistic expression E PTS: 1 TOP: Significance test for p1-p2: conclusion given P-value A PTS: 1 TOP: Significance test for p1-p2: test statistic expression A PTS: 1 TOP: Conclusion—association not causation C PTS: 1 TOP: Confidence interval for mu1-mu2 (formula) C PTS: 1 TOP: Significance test for mu1-mu2: calculate test statistic B PTS: 1 TOP: Significance test of mu1-mu2: conclude from Conf. Int. D PTS: 1 TOP: Confidence interval for mu1-mu2 (formula) E PTS: 1 TOP: Conditions for inference mu1-mu2 B PTS: 1 Conservative versus software values for degrees of freedom E PTS: 1 TOP: Confidence interval for mu1-mu2 (computation) E PTS: 1 TOP: Conclusion—given P-value D PTS: 1 TOP: Confidence interval for mu1-mu2 (computation) B PTS: 1 TOP: Interpret confidence interval D PTS: 1 TOP: Interpret P-value B PTS: 1 Conservative versus software values for degrees of freedom C PTS: 1 TOP: Paired vs. independent samples E PTS: 1 TOP: Paired vs. independent samples
1
