The U.S. Bureau of Labor Statistics (BLS) reported that the mean annual number of hours of vacation time earned by blue-collar and service employees who work for small private establishments and have at least 10 years of service is 100. Assume that for this population the standard deviation for the annual number of vacation hours earned is 48. Suppose the BLS would like to select a sample of 15,000 individuals from this population for a follow-up study. 

a. Show the sampling distribution of x̅, the sample mean for a sample of 15,000 individuals from this population. 

b. What is the probability that a simple random sample of 15,000 individuals from this population will provide a sample mean that is within one hour of the population mean? 

c. Suppose the mean annual number of hours of vacation time earned for a sample of 15,000 blue-collar and service employees who work for small private establishments and have at least 10 years of service differs from the population mean μ by more than one hour. Considering your results for part (b), how would you interpret this result?