Hayes (2000) took a stratified sample of New York City food stores. The sampling frame consisted of 1408 food stores with at least 4000 square feet of retail space. the population of stores was stratified into three strata using median household income within the zip code. The prices of a "market basket" of goods were determined for each store; the goal of the survey was to investigate whether prices differ among the three strata. Hayes used the logarithm of the total price for the basket as the response y. Results are given in the following table.
Stratum, h Nh nh ¯yh sh
1 Low income 190 21 3.925 0.037
2 Middle income 407 14 3.938 0.052
3 Upper income 811 22 3.942 0.070
(a) The planned sample size was 30 in each stratum; this was not achieved because some stores went out of business while the data were being collected. What are the advantages and disadvantages of sampling the same number of stores in each stratum?
(b) Estimate y¯U for these data and give a 95% CI.
(c) How would you allocate the sample sizes among three strata? Why?