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 \(\$ 36,961\). A sample of 1,000 females was randomly chosen from the entire population of the United States 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.

b. Would the probability be larger, or smaller, if the standard deviation of all females' incomes was \(\$ 25,000\) ? Why?