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The article "Psychiatric and Alcoholic Admissions Do Not Occur Disproportionately Close to Patients' Birthdays" (Psychological Reports. 1092: 944-946) focuses on the existence of any relationship between the date of patient admission for treatment of alcoholism and the patient's birthday. Assuming a 365-day year (i.e., excluding leap year), in the absence of any relation, a patient's admission date is equally likely to be any one of the 365 possible days. The investigators established four different admission categories: (1) within 7 days of birthday: (2) between 8 and 30 days, inclusive, from the birthday: (3) between 31 and 90 days, inclusive, from the birthday; and (4) more than 90 days from the birthday. A sample of 200 patients gave observed frequencies of 11, 24, 60. and 06 for categories 1, 2. 3, and 4, respectively. State and test the relevant hypotheses using a significance level of .01.If a process variable is normally distributed, in the long run virtually all observed values should be between u - 30 and u + 30 , giving a process spread of 60. a. With LSL and USL denoting the lower and upper specification limits, one commonly used process capability index is u - 30 . The value Co = 1 indicates a process that is only marginally capable of meeting specifications. Ideally, C. should exceed 1.33 (a "very good" process). Calculate the value of C, for each of the cork production processes described in the previous exercise, and comment. b. The Co index described in (a) does not take into account process location. A capability measure that does involve the process mean is M + 30. Calculate the value of Cox for each of the cork-production processes described in the previous exercise, and comment. [Vote: In practice, m and s have to be estimated from process data; we show how to do this in Section 16.2] c. How do Go and Gok compare, and when are they equal?Some sources advocate a somewhat more restrictive type of doubling-sampling plan in which r = G + 1; that is, the lot is rejected if at either stage the (total)) number of defectives is at least r (see the book by Montgomery). Consider this type of sampling plan with n, = 50, n, = 100, G, = 1, and r, = 4. Calculate the probability of lot acceptance when p = _02, 05, and .10.