4. A county government is interested in expanding the facilities of a day-care centre for mentally retarded children. The expansion will increase the cost of enrolling a child in the centre. A sample survey will be conducted to estimate the proportion of families with retarded children that will make use of the expanded facilities. The families are divided into those who use the existing facilities and those who do not. Some families live in the city in which the centre is located, and some live in the surrounding suburban and rural areas. Thus, stratied random sampling is used, with users in the city, users in the surrounding country, nonusers in the city, and nonusers in the country forming strata 1, 2, 3, and 4, respectively. Approximately 90% of the present users and 50% of the present nonusers will use the expanded facilities. The cost of obtaining an observation from a user is $4 and from a nonuser is $8. The difference in cost results because nonusers are difficult to locate. Existing records give N1 = 97, N2 = 4-3, N3 145, and N4 = 68. a) which method should be appropriate for this estimation? Find the approximate sample size based on the most appropriate method of allocation to estimate the population proportion with a bound of 0.05 on the error of estimation. b) The survey is conducted and the proportion of families who will use the new facilities are as follows: 331 = 0.87, 332 = 0.93, pa = 0.60, 334 = 0.53 Estimate the population proportion, 395:, and place a bound on the error of estimation. Was the desired bound in a) achieved? c) Suppose the total cost of sampling is fixed at $400. Choose the sample size and allocation that minimize the variance of the estimator pst for this fixed cost