The owners of Aunt Erma’s Restaurant 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 randomly sample three people in Boston. 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. Identify 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. Explain why the sampling distribution of the sample proportion who prefer Aunt Erma’s pizza, when n = 3, is
Sample Proportion Probability
0.......... 1/8
1/3.......... 3/8
2/3.......... 3/8
1.......... 1/8
c. In part b, we can also find the probabilities for each possible sample proportion value using the binomial distribution. Use the binomial with n = 3 and p = 0.50 to show that the probability of a sample proportion of 1/3 equals 3/8. (This equals the probability that x = 1 person out of n = 3 prefer pizza A. It’s especially helpful to use the binomial formula when p differs from 0.50, since then the eight possible samples listed in part a would not be equally likely.)