A university located in a city wishes to estimate what proportion of the 1000 students in its 4 halls of residence regularly walk to the campus for their classes. The halls are at very different distances from the campus, so that the proportions are likely to differ between halls. The table below gives the number of students in each hall and a guess at the likely proportions of walkers. Hall A B C D Number of students 400 300 100 200 Guess at proportion of walkers 0.9 0.8 0.5 0.2 It is decided to take a sample of 100 students using stratified random sampling with the halls as strata. (a) Defining any notation you use, explain how Neyman allocation would divide this sample of 100 between the 4 halls and calculate the numbers to be sampled from each hall under this scheme. In what circumstances is this the optimal allocation? [6] (b) Defining any further notation you use, write down the formula for the usual estimate of the population proportion when using stratified random sampling. You are not required to compute anything for this example. [2] (c) Assuming for this purpose that the guessed proportions are correct, use the data in the table above to calculate the variance of the estimator in (b) under Neyman allocation. [4] (d) Calculate the sample sizes for proportional allocation, compare them with those for Neyman allocation and comment on the differences. [4] (e) Using the data in the table above calculate the variance of the estimator in (b) under proportional allocation and comment on how it compares with the variance for Neyman allocation. [6]