A study was initiated having as a major objective estimation of the number of patients with severe trauma seen during the calendar year 2004 in hospitals not designated as trauma centers in urban counties in Illinois outside of the Chicago area which have both Trauma Centers (T) and hospitals not designated as Trauma Centers (NT). The following data were available to the investigators.

Use a ratio estimate based on number of beds to estimate from the above sample the total number of cases of severe trauma among hospitals in these 20 counties not designated as trauma centers.
12 A sample survey of patient records is being planned in a city that has 25 local mental health centers. The objective of the survey is to estimate the total number of patients who were given diazepam (Valium) as part of their therapeutic regimen. The number of patients treated in each of the mental health centers is listed in the accompanying table. A sample of patients is to be selected by choosing a simple random sample of mental health centers and, within each sample mental health center, by choosing a subsample of patients.

a. Based on the data for these five health centers, calculate the intraclass correlation coefficient with respect to the number of patients given diazepam.

b. Based on the above calculated intraclass correlation coefficient, what would be the optimal average cluster size n for the proposed survey of the 25 health centers listed above if the cost component associated with clusters is one thousand times that associated with listing units?