(c) Repeat step (b) 20 times to get 20 resamples and 20 values of the statis-tic. Make a histogram of the distribution of these 20 values. This is the permutation distribution for your resamples.
(d) What proportion of the 20 statistic values were equal to or greater than the original value in part (a)? You have just estimated the one-sided P-value for the original 6 observations.
14.46 Table 14.1 contains the selling prices for a random sample of 50 Seattle real estate transactions in 2002. Table 14.5 contains a similar random sample of sales in 2001. Test whether the means of the two random samples of the 2001 and 2002 real estate sales data are significantly different.
TA B L E 1 4 . 5
Sellingprices for Seattle real estate, 2001 ($1000s)
419 55.268 65 210 510.728 212.2 152.720 266.6 69.427 125
191 451 469 310 325 50 675 140 105.5 285
320 305 255 95.179 346 199 450 280 205.5 135
190 452.5 335 455 291.905 239.9 369.95 569 481 475
495 195 237.5 143 218.95 239 710 172 228.5 270
(a) State the null and alternative hypotheses.
(b) Perform a two-sample t test. What is the P-value?
(c) Perform a permutation test on the difference in means. What is the P-value? Compare it with the P-value you found in part (b). What do you conclude based on the tests?
(d) Find a bootstrap BCa 95% confidence interval for the difference in means.
How is the interval related to your conclusion in (c)?
14.47 Here are heights (inches) of professional female basketball players who are centers and forwards. We wonder if the two positions differ in average height.
Forwards
69 72 71 66 76 74 71 66 68 67 70 65 72
70 68 73 66 68 67 64 71 70 74 70 75 75
69 72 71 70 71 68 70 75 72 66 72 70 69
Centers
72 70 72 69 73 71 72 68 68 71 66 68 71
73 73 70 68 70 75 68
(a) Make a back-to-back stemplot of the data. How do the two distributions compare?
(b) State null and alternative hypotheses. Do a permutation test for the differ-ence in means of the two groups. Give the P-value and draw a conclusion.
14.48 A customer complains to the owner of an independent fast-food restaurant that the restaurant is discriminating against the elderly. The customer claims
that people 60 years old and older are given fewer french fries than people un-der 60. The owner responds by gathering data, collected without the knowl-edge of the employees so as not to affect their behavior. Here are data on the weight of french fries (grams) for the two groups of customers:
Age< 60: 75 77 80 69 73 76 78 74 75 81 Age≥ 60: 68 74 77 71 73 75 80 77 78 72
(a) Display the two data sets in a back-to-back stemplot. Do they appear sub-stantially different?
(b) If we perform a permutation test using the mean for “< 60” minus the mean for “≥ 60,” should the alternative hypothesis be two-sided, greater, or less? Explain.
(c) Perform a permutation test using the chosen alternative hypothesis and give the P-value. What should the owner report to the customer?
14.49 Verizon uses permutation testing for hundreds of comparisons, such as be-tween different time periods, bebe-tween different locations, and so on. Here is a sample from another Verizon data set, containing repair times in hours for Verizon (ILEC) and CLEC customers.
ILEC
1 1 1 1 2 2 1 1 1 1 2 2 1 1 1 1
2 2 1 1 1 1 2 3 1 1 1 1 2 3 1 1
1 1 2 3 1 1 1 1 2 3 1 1 1 1 2 3
1 1 1 1 2 3 1 1 1 1 2 4 1 1 1 1
2 5 1 1 1 1 2 5 1 1 1 1 2 6 1 1
1 1 2 8 1 1 1 1 2 15 1 1 1 2 2
CLEC
1 1 5 5 5 1 5 5 5 5
(a) Choose and make data displays. Describe the shapes of the samples and how they differ.
(b) Perform a t test to compare the population mean repair times. Give hy-potheses, the test statistic, and the P-value.
(c) Perform a permutation test for the same hypotheses using the pooled-variance t statistic. Why do the two P-values differ?
(d) What does the permutation test P-value tell you?
14.50 The estimated P-value for the DRP study (Example 14.12) based on 999 re-samples is P = 0.015. For the Verizon study (Example 14.13) the estimated P-value for the test based on x1− x2 is P = 0.0183 based on 500,000 re-samples. What is the approximate standard deviation of each of these es-timated P-values? (Use each P as an estimate of the unknown true P-value p.) 14.51 You want to test the equality of the means of two populations. Sketch density
CHALLENGE
curves for two populations for which
(a) a permutation test is valid but a t test is not.
(b) both permutation and t tests are valid.
(c) a t test is valid but a permutation test is not.
Exercises 14.52 to 14.63 concern permutation tests for hypotheses stated in terms of a variety of parameters. In some cases, there are no standard formula-based tests for the hypotheses in question. These exercises illustrate the flexibility of permutation tests.
14.52 Because distributions of real estate prices are typically strongly skewed, we of-ten prefer the median to the mean as a measure of center. We would like to test the null hypothesis that Seattle real estate sales prices in 2001 and 2002 have equal medians. Sample data for these years appear in Tables 14.1 and 14.5.
Carry out a permutation test for the difference in medians, find the P-value, and explain what the P-value tells us.
14.53 Exercise 7.41 (page 482) gives data on a study of the effect of a summer lan-guage institute on the ability of high school lanlan-guage teachers to understand spoken French. This is a matched pairs study, with scores for 20 teachers at the beginning (pretest) and end (posttest) of the institute. We conjecture that the posttest scores are higher on the average.
(a) Carry out the matched pairs t test. That is, state hypotheses, calculate the test statistic, and give its P-value.
(b) Make a normal quantile plot of the gains: posttest score− pretest score.
The data have a number of ties and a low outlier. A permutation test can help check the t test result.
(c) Carry out the permutation test for the difference of means in matched pairs, using 9999 resamples. The normal quantile plot shows that the permuta-tion distribupermuta-tion is reasonably normal, but the histogram looks a bit odd.
What explains the appearance of the histogram? What is the P-value for the permutation test? Do your tests in (a) and (c) lead to the same practi-cal conclusion?
14.54 Table 14.2 contains the salaries and batting averages of a random sample of 50 Major League Baseball players. Can we conclude that the correlation between these variables is greater than 0 in the population of all players?
(a) State the null and alternative hypotheses.
(b) Perform a permutation test based on the sample correlation. Report the P-value and draw a conclusion.
14.55 In Exercise 14.39, we assessed the significance of the correlation between re-turns on Treasury bills and common stocks by creating bootstrap confidence intervals. If a 95% confidence interval does not cover 0, the observed corre-lation is significantly different from 0 at theα = 0.05 level. We would prefer to do a test that gives us a P-value. Carry out a permutation test and give the P-value. What do you conclude? Is your conclusion consistent with your work in Exercise 14.39?
14.56 The formal medical term for vitamin A in the blood is serum retinol. Serum retinol has various beneficial effects, such as protecting against fractures.
Medical researchers working with children in Papua New Guinea asked whether recent infections reduce the level of serum retinol. They classified children as recently infected or not on the basis of other blood tests, then measured serum retinol. Of the 90 children in the sample, 55 had been re-cently infected. Table 14.6 gives the serum retinol levels for both groups, in micromoles per liter.13
TA B L E 1 4 . 6
Serum retinol levels in two groups of children
Not infected Infected
0.59 1.08 0.88 0.62 0.46 0.39 0.68 0.56 1.19 0.41 0.84 0.37
1.44 1.04 0.67 0.86 0.90 0.70 0.38 0.34 0.97 1.20 0.35 0.87
0.35 0.99 1.22 1.15 1.13 0.67 0.30 1.15 0.38 0.34 0.33 0.26
0.99 0.35 0.94 1.00 1.02 1.11 0.82 0.81 0.56 1.13 1.90 0.42
0.83 0.35 0.67 0.31 0.58 1.36 0.78 0.68 0.69 1.09 1.06 1.23
1.17 0.35 0.23 0.34 0.49 0.69 0.57 0.82 0.59 0.24 0.41
0.36 0.36 0.39 0.97 0.40 0.40 0.24 0.67 0.40 0.55 0.67 0.52 0.23 0.33 0.38 0.33 0.31 0.35 0.82
(a) The researchers are interested in the proportional reduction in serum retinol. Verify that the mean for infected children is 0.620 and that the mean for uninfected children is 0.778.
(b) There is no standard test for the null hypothesis that the ratio of the pop-ulation means is 1. We can do a permutation test on the ratio of sample means. Carry out a one-sided test and report the P-value. Briefly describe the center and shape of the permutation distribution. Why do you expect the center to be close to 1?
14.57 In Exercise 14.56, we did a permutation test for the hypothesis “no difference
CHALLENG
E between infected and uninfected children” using the ratio of mean serum retinol levels to measure “difference.” We might also want a bootstrap confi-dence interval for the ratio of population means for infected and uninfected children. Describe carefully how resampling is done for the permutation test and for the bootstrap, paying attention to the difference between the two resampling methods.
14.58 Here is one conclusion from the data in Table 14.6, described in Exercise 14.56: “The mean serum retinol level in uninfected children was 1.255 times the mean level in the infected children. A 95% confidence interval for the ratio of means in the population of all children in Papua New Guinea is. . . .”
(a) Bootstrap the data and use the BCa method to complete this conclusion.
(b) Briefly describe the shape and bias of the bootstrap distribution. Does the bootstrap percentile interval agree closely with the BCa interval for these data?
14.59 In Exercise 14.49 we compared the mean repair times for Verizon (ILEC) and CLEC customers. We might also wish to compare the variability of repair times. For the data in Exercise 14.49, the F statistic for comparing sample variances is 0.869 and the corresponding P-value is 0.67. We know that this test is very sensitive to lack of normality.
(a) Perform a two-sided permutation test on the ratio of standard deviations.
What is the P-value and what does it tell you?
(b) What does a comparison of the two P-values say about the validity of the F test for these data?
14.60 Does added calcium intake reduce the blood pressure of black men? In a ran-domized comparative double-blind trial, 10 men were given a calcium sup-plement for twelve weeks and 11 others received a placebo. For each subject, record whether or not blood pressure dropped. Here are the data:14
Treatment Subjects Successes Proportion
Calcium 10 6 0.60
Placebo 11 4 0.36
Total 21 10 0.48
We want to use these sample data to test equality of the population proportions of successes. Carry out a permutation test. Describe the permutation distri-bution. The permutation test does not depend on a “nice” distribution shape.
Give the P-value and report your conclusion.
14.61 We want a 95% confidence interval for the difference in the proportions of reduced blood pressure between a population of black men given calcium and a similar population given a placebo. Summary data appear in Exercise 14.60.
(a) Give the plus four confidence interval. Because the sample sizes are both small, we may wish to use the bootstrap to check this interval.
(b) Bootstrap the sample data. We recommend tilting confidence intervals for proportions based on small samples. Other bootstrap intervals may be in-accurate. Give all four bootstrap confidence intervals (t, percentile, BCa, tilting). How do the other three compare with tilting? How does the tilting interval compare with the plus four interval?
14.62 We prefer measured data to the success/failure data given in Exercise 14.60.
Table 14.7 gives the actual values of seated systolic blood pressure for this experiment. Example 7.20 (page 501) applies the pooled t test (a procedure that we do not recommend) to these data. Carry out a permutation test to dis-cover whether the calcium group had a significantly greater decrease in blood pressure.
TA B L E 1 4 . 7
Effect of calcium and placebo on blood pressure
Calcium group Placebo group
Begin End Decrease Begin End Decrease
107 100 7 123 124 −1
110 114 −4 109 97 12
123 105 18 112 113 −1
129 112 17 102 105 −3
112 115 −3 98 95 3
111 116 −5 114 119 −5
107 106 1 119 114 5
112 102 10 114 112 2
136 125 11 110 121 −11
102 104 −2 117 118 −1
130 133 −3
14.63 Are the variances of decreases in blood pressure equal in populations of black men given calcium and given a placebo? Example 7.22 (page 518) applied the F test for equality of variances to the data in Table 14.7. This test is unreliable because it is sensitive to nonnormality in the data. The permutation test does not suffer from this drawback.
(a) State the null and alternative hypotheses.
(b) Perform a permutation test using the F statistic (ratio of sample vari-ances) as your statistic. What do you conclude?
(c) Compare the permutation test P-value with that in Example 7.22. What do you conclude about the F test for equality of variances for these data?
14.64 Exercise 7.27 (page 478) gives these data on a delicate measurement of total
CHALLENGE
body bone mineral content made by two operators on the same 8 subjects:15 Subject
Operator 1 2 3 4 5 6 7 8
1 1.328 1.342 1.075 1.228 0.939 1.004 1.178 1.286 2 1.323 1.322 1.073 1.233 0.934 1.019 1.184 1.304 Do permutation tests give good evidence that measurements made by the two operators differ systematically? If so, in what way do they differ? Do two tests, one that compares centers and one that compares spreads.