Suppose a boxplot of the differences in the pairs from a paired data set has many out-

Bio. 6.36 The effect of Benzedrine on the heart rate of dogs in beats per minute was examined in an experiment on 14 dogs chosen for the study. Each dog was to serve as its own control, with half of the dogs assigned to receive Benzedrine during the first study period and the other half assigned to receive a placebo saline solution. All dogs were examined to determine the heart rates after 2 hours on the medication. After 2 weeks in which no medication was given, the regimens for the dogs were switched for the second study period. The dogs previously on Benzedrine were given the placebo and the others received Benzedrine. Again heart rates were measured after 2 hours. The following sample data are not arranged in the order in which they were taken but have been summarized by regimen. Use these data to test the research hypothesis that the distribution of heart rates for the dogs when receiving Benzedrine is shifted to the right of that for the same animals when on the placebo. Use a one-tailed Wilcoxon signed-rank test with a ⫽ .05. Dog Placebo Benzedrine Dog Placebo Benzedrine 1 250 258 8 296 305 2 271 285 9 301 319 3 243 245 10 298 308 4 252 250 11 310 320 5 266 268 12 286 293 6 272 278 13 306 305 7 293 280 14 309 313 6.6 Choosing Sample Sizes for Inferences about M 1 ⴚ M 2 Med. 6.37 A study is being planned to evaluate the possible side effects of an anti-inflammatory drug. It is suspected that the drug may lead to an elevation in the blood pressure of users of the drug. A pre- liminary study of two groups of patients, one receiving the drug and the other receiving a placebo, provides the following information on the systolic blood pressure in mm Hg of the two groups: Group Mean Standard Deviation Placebo 129.9 18.5 Users of drug 135.5 18.7 Assume that both groups have systolic blood pressures that have a normal distribution with stan- dard deviations relatively close to the values obtained in the pilot study. Suppose the study plan provides for the same number of patients in the placebo as in the treatment group. Determine the sample size necessary for an a ⫽ .05 t test to have power of .80 to detect an increase of 5 mm Hg in the blood pressure of the treatment group relative to the placebo group. Med. 6.38 Refer to Exercise 6.37. Suppose that the agency sponsoring the study specifies that the group receiving the drug should have twice as many patients as the placebo group. Determine the sample sizes necessary for an a ⫽ .05 t test to have power of .80 to detect an increase of 5 mm Hg in the blood pressure of the treatment group relative to the placebo group. Med. 6.39 Refer to Exercise 6.37. The researchers also need to obtain precise estimates of the mean difference in systolic blood pressure for people who use the anti-inflammatory drug versus those who do not.

a. Suppose the sample sizes are the same for both groups. What sample size is needed

to obtain a 95 confidence interval for the mean difference in systolic blood pressure between the users and nonusers having a width of at most 5 mm Hg.

b. Suppose the user group will have twice as many patients as the placebo group. What

sample size is needed to obtain a 95 confidence interval for the mean difference in sys- tolic blood pressure between the users and nonusers having a width of at most 5 mm Hg. Env. 6.40 An environmental impact study was performed in a small state to determine the effective- ness of scrubbers on the amount of pollution coming from the cooling towers of a chemical plant. The amount of pollution in ppm detected from the cooling towers before and after the scrub- bers were installed is given below for 23 chemical plants. Mean Standard Deviation Before Scrubber 71 26 After Scrubber 63 25 Difference ⫽ Before ⫺ After 8 20 Suppose a larger study is planned for a state with a more extreme pollution problem.

a. How many chemical plant cooling towers need to be measured if we want a probabil-

ity of .90 of detecting a mean reduction in pollution of 10 ppm due to installing the scrubber using an a ⫽ .01 test? b. What assumptions did you make in part a in order to compute the sample size? Env. 6.41 Refer to Exercise 6.40. The state regulators also need to obtain precise estimates of the mean reduction in the pollution levels after installing the scrubber. What sample size is needed to obtain a 95 confidence interval for the mean reduction using a 99 confidence interval having width of 10 ppm? Supplementary Exercises Med. 6.42 Long-distance runners have contended that moderate exposure to ozone increases lung capacity. To investigate this possibility, a researcher exposed 12 rats to ozone at the rate of 2 parts per million for a period of 30 days. The lung capacity of the rats was determined at the beginning of the study and again after the 30 days of ozone exposure. The lung capacities in mL are given here. Rat 1 2 3 4 5 6 7 8 9 10 11 12 Before exposure 8.7 7.9 8.3 8.4 9.2 9.1 8.2 8.1 8.9 8.2 8.9 7.5 After exposure 9.4 9.8 9.9 10.3 8.9 8.8 9.8 8.2 9.4 9.9 12.2 9.3

a. Is there sufficient evidence to support the conjecture that ozone exposure increases

lung capacity? Use a ⫽ .05. Report the p-value of your test.

b. Estimate the size of the increase in lung capacity after exposure to ozone using a 95

confidence interval.

c. After completion of the study, the researcher claimed that ozone causes increased

lung capacity. Is this statement supported by this experiment? Env. 6.43 In an environmental impact study for a new airport, the noise level of various jets was measured just seconds after their wheels left the ground. The jets were either wide-bodied or narrow-bodied. The noise levels in decibels dB are recorded here for 15 wide-bodied jets and 12 narrow-bodied jets. Wide-Bodied Jet 109.5 107.3 105.0 117.3 105.4 113.7 121.7 109.2 108.1 106.4 104.6 110.5 110.9 111.0 112.4 Narrow-Bodied Jet 131.4 126.8 114.1 126.9 108.2 122.0 106.9 116.3 115.5 111.6 124.5 116.2 a. Do the two types of jets have different mean noise levels? Report the level of significance of the test.

b. Estimate the size of the difference in mean noise level between the two types of jets

using a 95 confidence interval. c. How would you select the jets for inclusion in this study? Ag.

6.44 An entomologist is investigating which of two fumigants, F

1 or F 2 , is more effective in controlling parasities in tobacco plants. To compare the fumigants, nine fields of differing soil