7.49 Sampling distribution for small and large n The owners of Aunt Erma’s Restaurant in Boston plan an advertising campaign with the claim that more people prefer the taste of their pizza (which we’ll denote by A) than the current leading fast-food chain selling pizza (which we’ll denote by D). To support their claim, they plan to sample three people in Boston randomly. Each person is asked to taste a slice of pizza A and a slice of pizza D. Subjects are blindfolded so they cannot see the pizza when they taste it, and the order of giving them the two slices is randomized. They are then asked which pizza tastes better. Use a symbol with three letters to represent the responses for each possible sample. For instance, ADD represents a sample in which the first subject sampled preferred pizza A and the second and third subjects preferred pizza D.

a. List the eight possible samples of size 3, and for each sample report the proportion that preferred pizza A.

b. In the entire Boston population, suppose that exactly half would prefer pizza A and half would prefer pizza D. Then, each of the eight possible samples is equally likely to be observed. Explain why the sampling distribution of the sample proportion who prefer Aunt Erma’s pizza, when n = 3, is

c. In theory, you could use the same principle as in part b to find the sampling distribution for any n, but it is tedious to list all elements of the sample space. For instance, for n = 50, there are more than 1015 elements to list. Despite this, what is the mean, standard deviation, and approximate shape of the sampling distribution of the sample proportion when n = 50 (still assuming that the population proportion preferring pizza A is 0.5)?

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Sample Proportion Probability 0 1/8 1/3 3/8 2/3 3/8 1 1/8