Question

According to a social media​ blog, time spent on a certain social networking website has a mean of 24 minutes per visit. Assume that time spent on the social networking site per visit is normally distributed and that the standard deviation is 7 minutes.

a. If you select a random sample of 25 ​sessions, what is the probability that the sample mean is between 23.5 and 24.5 ​minutes? ​(Round to three decimal places as​ needed.)

b. If you select a random sample of 25 ​sessions, what is the probability that the sample mean is between 23 and 24 ​minutes? ​(Round to three decimal places as​ needed.)

c. If you select a random sample of 100 ​sessions, what is the probability that the sample mean is between 23.5 and 24.5 ​minutes? ​(Round to three decimal places as​ needed.)

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

The sample size in​ (c) is greater than the sample size in​ (a), so the standard error of the mean​ (or the standard deviation of the sampling​ distribution) in​ (c) is____________________ (less/greater) than in​ (a). As the standard error ____________________ (increases/decreases) values become more concentrated around the mean.​ Therefore, the probability that the sample mean will fall in a region that includes the population mean will always ____________________ (decrease/increase) when the sample size increases.

Answer 1

Here it is given that distribution is normal and also population standard deviation is known, so sample mean is also normally distributed

a. Here we need to find

b.

c.

d. The sample size in​ (c) is greater than the sample size in​ (a), so the standard error of the mean​ (or the standard deviation of the sampling​ distribution) in​ (c) is less than in​ (a). As the standard error decreases values become more concentrated around the mean.​ Therefore, the probability that the sample mean will fall in a region that includes the population mean will always increase when the sample size increases.