In a large state, there are 24 counties that are not in metropolitan areas. For purposes of estimating the total number of admissions during the calendar year 2000 to hospitals in this 24-county area with acquired immune deficiency syndrome (AIDS), these counties are grouped into six strata, and a simple random sample of two counties is selected within each stratum. Within each of the sample counties, a simple random sample of one hospital is taken, and within each sample hospital, all hospital medical records of patients having a discharge diagnosis of AIDS are reviewed, abstracted, and enumerated. The following table lists the hospitals by stratum, county, and number of beds: 

a. Show how you would estimate the number of AIDS admissions for the 24-
county area, taking into consideration the number of beds.

b. The hospitals actually sampled are shown in the accompanying table along with the number of AIDS admissions enumerated at each sample hospital.
From these data, estimate the total number of AIDS admissions for the 24-
county area.

c. Use a set of balanced half samples to estimate the standard error of the estimated total number of AIDS admissions for the 24-county area. The following Plackett-Burman matrix can be used to construct the half sample estimates (McCarthy |6, p. 17|):

1 1 72 2223 1 87 2 104 3 34 1 17 3 2 140 1 4 104 2 I 44 2233222 2 99 3 1 86 3 2 91 3 3 53 4 1 48 1 171 2 85 3 2 1 108 3 3 1 99 3 4 1 131 3 4 2 182 4 1 48 1 2 18 4 2 1 50 4 3 1 42 4 4 38