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Question 6 A company that delivers weekly food boxes to people's homes is considering adding an extra delivery route that will visit 4 villages of varying sizes. To gauge demand they propose to carry out a survey using stratified random sampling to select a number of addresses in each village and sending an interviewer to these addresses to gather data that will result in an estimate of the possible weekly spend on the company's food boxes by that household. The table below shows for each village the numbers of addresses the village, the sampling costs per address visited (the two larger ones are more spread out) and a guess at the within-village standard deviation of weekly spend. Page 3 of 4 (a) For a fixed total sample size \( n \) determine the optimal allocation of the sample between villages, taking account of sizes, costs and standard deviations. Define carefully the notation in any formula you quote. [4] (b) After allowing for all the other costs of the exercise, the budget available to cover the visits is £800. Use this budget to determine the total sample size \( n \) and the sample sizes for the 4 villages. [5] (c) Calculate the sample sizes for each village using proportional allocation with the same budget of £800. [7] (d) Carefully defining any notation you use, and without assuming which of the allocations in (b) or (c) will be used, write down expressions for the estimator of the mean weekly spend per household
(a) For a fixed total sample size \( n \) determine the optimal allocation of the sample between villages, taking account of sizes, costs and standard deviations. Define carefully the notation in any formula you quote. [4] (b) After allowing for all the other costs of the exercise, the budget available to cover the visits is £800. Use this budget to determine the total sample size \( n \) and the sample sizes for the 4 villages. [5] (c) Calculate the sample sizes for each village using proportional allocation with the same budget of £800. [7] (d) Carefully defining any notation you use, and without assuming which of the allocations in (b) or (c) will be used, write down expressions for the estimator of the mean weekly spend per household of the population 1650 houses in these villages and for its standard error. [5] (e) Use the formula you wrote down in (d) to compute the standard error of the estimator if the optimal allocation in (b) is used. [5] (f) Without making any further calculations, say whether you would expect this standard error to increase or decrease if proportional allocation were to be used. Explain how the better of the two allocations achieves its improvement. Is this improvement likely to be a large one? [5]

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