6. Researchers at a large health maintenance organization (HMO) are planning a study of a certain mild illness. They will select a random sample of patients who are ages 35 to 54 and see if they contract the illness in the next year. The researchers are interested in estimating the proportions of men and of women who are likely to develop the illness in each of 4 agegroups: 3539, 4044, 4549, and 5054. The researchers plan to include 2,000 patients in the study. Suppose the researchers draw a random sample from all of the patients at this HMO who are ages 35 to 54 and nd the following numbers within each gender and age-group. AgeGroup 3539 40-44 45-49 50-54 Male 350 230 150 60 Female 445 370 245 150 (a) _--- .u. ....... . .- .uv. '.-- 'vv...- \\. v.... ' Suppose that at the end of the study, 10 percent of the females in the 4044 agegroup contracted the illness. Calculate a 95 percent condence interval to estimate the population proportion of females in this agegroup that contracted the illness. Interpret this condence interval in the context of this situation. Interpret the condence level of 95 percent. (b) Suppose that at the end of the study, 10 percent of the males in the 4044 agegroup contracted the illness. The corresponding 95 percent condence interval to estimate the population proportion of males in this age group that contracted the illness is (0.061, 0.139). Note that this interval and the interval in part (a) are of different lengths even though the two sample proportions were identical. What would be an alternative way to allocate a sample of 2,000 subjects so that the 95 percent condence interval widths for all male agegroups and for all female agegroups (i.e., for all 8 groups) would be the same when the sample proportions are the same? Justify your answer. Based on previous studies, researchers believe that the percentages of those who contract the illness will be similar for males and females, and therefore plan to ignore gender when selecting a sample for this study. Previous studies also indicate that the percentages of adults who will contract this illness in the 3539, 4044, 4549, and 5054 agegroups are anticipated to be 5%, 8%, 20%, and 35%, respectively. How should the sample of 2,000 subjects be allocated with respect to age-groups so that the widths of the 95 percent condence intervals for the four groups will be approximately the same? Justify your