Chapter 7.6, Problem 32E

The Wall Street Journal reported that the age at first startup for 55% of entrepreneurs was 29 years or less and the age at first startup for 45% of entrepreneurs was 30 years or more.
a. Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of $\bar{p}$ where $\bar{p}$ is the sample proportion of entrepreneurs whose first startup was at 29 years of age or less.
b. What is the probability that the sample proportion in part (a) will be within $\pm .05$ of its population proportion?
c. Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of $\bar{p}$ where $\bar{p}$ is now the sample proportion of entrepreneurs whose first startup was at 30 years of age or more.
d. What is the probability that the sample proportion in part (c) will be within $\pm .05$ of its population proportion?
e. Is the probability different in parts (b) and (d)? Why?
f. Answer part (b) for a sample of size 400. Is the probability smaller? Why?