Another set of alternatives is:
Twenty quail were randomly assigned to two groups, 10 to each. The quail in Group I were given a diet without a drug compound while the quail in Group II were given a diet with a drug compound inserted, which hopefully reduces LDL (low-density-lipid) cholesterol. Except for the difference in diet the quail were treated the same. At the end of the study their LDL levels were measured. The hypotheses are:
Group I: 47 52 54 67 68 69 73 79 116 120 Group II: 30 30 31 33 34 36 47 59 98 125and a dotplot
. .. :. . . . . Group I +---------+---------+---------+---------+---------+------- :.:. . . . . Group II +---------+---------+---------+---------+---------+------- 20 40 60 80 100 120It appears that the drug compound had an effect. The value of the Wilcoxon is T = 21.5 which is certainly smaller than what we would expect if H_{0} is true, i.e., . Is this small enough? We need the p-value which is the probability that under H_{0}. To estimate this, we obtained 100 resampled T 's:
22.0 23.5 24.5 28.0 28.0 30.0 30.5 30.5 31.0 31.0 32.5 35.0 36.5 37.0 37.0 37.5 38.5 39.0 39.0 39.5 40.0 40.5 41.0 41.5 41.5 41.5 42.0 43.0 43.0 43.5 44.0 44.0 44.5 45.5 46.0 46.5 46.5 46.5 46.5 47.5 48.0 48.0 48.0 48.5 48.5 49.0 49.0 49.0 50.5 50.5 50.5 50.5 51.0 51.0 51.0 51.0 51.5 51.5 52.0 52.5 52.5 52.5 53.0 53.0 53.5 54.0 54.0 54.5 55.0 55.0 55.0 55.5 56.0 56.5 56.5 57.5 57.5 57.5 58.5 59.5 59.5 59.5 61.0 61.5 61.5 62.0 64.0 64.5 64.5 66.0 66.0 66.0 66.5 68.0 69.0 71.0 71.0 72.0 74.0 75.5The estimated p-value is 0/100 = 0. Here is a picture of the p-value
: . : . : . . : . ::: :.:.: . . . . . .. : .: . . :.::::::.:::: ::::::::.:.: .::... :. . . +X--------+---------+---------+---------+---------+------- 20 30 40 50 60 70I also ran 1000 resampled T's which resulted in 13 resampled T's being less than or equal to 21.5. Hence the p-value is 13/1000 = .013. Based on this evidence, we reject H_{0} in favor of H_{A} and conclude that the drug is effective in reducing LDL cholesterol.
The third alternative is the alternative of ignorance; i.e., the populations differ. Formally,
From Statistical Concepts and Methods, Page 321, Bhattacharya and Johnson (1977), New York: Wiley. The peak oxygen intake per unit of body weight, called the aerobic capacity of an individual performing a strenuous activity is a measure of work capacity. For a comparative study, measurements are recorded for a group of 12 Peruvian Highland natives and 10 U.S. lowlanders who have spent considerable time in high altitudes. Do these groups seem to differ in peak oxygen intake? The hypotheses are:
Peru 34 35 36 38 38 42 43 46 48 50 52 55 US 30 32 32 33 36 38 41 43 44 46Let T be the number of times a Peruvian has a higher peak oxygen intake than a US person. The value of the Wilcoxon is T = 77.5 which exceeds 120/2 = 60. So T is on the upside. Hence the p-value is twice the probability that T is greater than or equal to 87.5. To estimate the p-value here are 100 resampled T's under H_{0}:
24.5 24.5 31.5 32.5 34.0 35.0 38.0 38.5 39.5 41.5 42.0 42.5 44.0 44.5 45.0 45.0 45.5 46.5 47.0 47.5 48.5 48.5 49.5 51.0 52.0 52.0 52.0 52.5 52.5 52.5 52.5 53.0 53.0 53.5 54.0 54.5 54.5 54.5 54.5 55.0 55.5 56.0 57.0 57.0 57.5 58.5 60.5 60.5 61.5 61.5 62.0 62.5 62.5 63.0 63.5 63.5 63.5 64.0 64.5 65.0 65.0 65.5 65.5 65.5 65.5 66.0 67.0 68.0 68.0 68.5 68.5 69.0 69.5 69.5 69.5 70.0 70.0 70.5 70.5 71.0 73.0 74.0 75.0 76.5 77.0 77.0 77.5 78.5 79.0 80.5 83.5 85.0 85.0 86.5 89.0 89.0 89.5 90.5 93.5 101.0Based on these resampled T's, we estimate the p-value to be 2*6/100 = .12. Assuming this pattern holds for 1000 resampled T's, we would not reject H_{0} in favor of H_{A}. Our conclusion would be, that there is insufficient evidence that Peruvian Highlanders differ from U.S. acclimatized Lowlanders with reference to peak oxygen intake. You are asked in the problems to estimate the p-value based on 1000 samples.
A: 12 16 18 25 30 B: 8 10 19 22 28
A: 70 72 87 88 102 112 B: 41 43 54 67 74 78 87 91
Next change, the 70 to 7, the first A. Determine the Wilcoxon test statistic and the p-value based on 1000 resamples using class code (Two-Sample Hypothesis and CI (Wilcoxon)). Did your conclusion change? Next change it to -7000. Determine the Wilcoxon test statistic and the p-value based on 1000 resamples using class code (Two-Sample Hypothesis and CI (Wilcoxon)). Did your conclusion change?
State H_{0} and H_{A}. Use the Wilcoxon to test these hypotheses. Use 1000 resampled T's. Obtain comparison dotplots. Conclude in terms of the problem.