According to a report by App Annie, a business intelligence company that produces tools and reports for the apps and digital goods industry, smartphone owners are using an average of 30 apps per month. 

Source: “Report: Smartphone owners are using 9 apps per day, 30 per month,” 2017, tcrn.ch/2qK4iRr.

Assume that number of apps used per month by smartphone owners is normally distributed and that the standard deviation is 5. If you select a random sample of 25 smartphone owners,

a. What is the probability that the sample mean is between 29 and 31?

b. What is the probability that the sample mean is between 28 and 32?

c. If you select a random sample of 100 smartphone owners, what is the probability that the sample mean is between 29 and 31?

d. Explain the difference in the results of (a) and (c).