Chapter 5, Problem 5.71ACI

Switching banks after a merger. Banks that merge with others to form “mega-banks” sometimes leave customers dissatisfied with the impersonal service. A poll by the Gallup Organization found 20% of retail customers switched banks after their banks merged with another. One year after the acquisition of First Fidelity by First Union, a random sample of 250 retail customers who had banked with First Fidelity were questioned. Let $\hat{p}$ be the proportion of those customers who switched their business from First Union to a different bank.

1. a. Find the mean and the standard deviation of $\hat{p}$.
2. b. Calculate the interval $E\left(\hat{p}\right)\pm 2{\sigma }_{\hat{p}}$.
3. c. If samples of size 250 were drawn repeatedly a large number of times and $\hat{p}$ determined for each sample, what proportion of the $\hat{p}$ values would fall within the interval you calculated in part c?