7.24 Survey accuracy According to the U.S. Census Bureau, Current Population Survey, Annual Social and Economic Supplement, the average income for females was $28,466 and the standard deviation was $36,961 in 2015. A sample of 1,000 females was randomly chosen from the entire United States population to verify if this sample would have a similar mean income as the entire population.

a. Find the probability that the mean income of the females sampled is within two thousand of the mean income for all females. (Hint: Find the sampling distribution of the sample mean income and use the central limit theorem).

b. Would the probability be larger or smaller if the standard deviation of all females’ incomes was $25,000?

Why?