According to the U.S. Department of Health and Human Services, only 49.2% of homes in the United States used landline telephone service in 2017. 

a. Suppose a sample of 207,000 U.S. homes will be taken to learn about home telephone usage. Show the sampling distribution of p̅ where p̅ is the sample proportion of homes that use landline phone service. 

b. What is the probability that the sample proportion in part (a) will be within ± .002 of the population proportion? 

c. Suppose a sample of 86,800 entrepreneurs will be taken to learn about home telephone usage. Show the sampling distribution of p̅ where p̅ is the sample proportion of homes that use landline phone service.

d. What is the probability that the sample proportion in part (c) will be within ± .002 of the population proportion? 

e. Are the probabilities different in parts (b) and (d)? Why or why not?