Question

The company Digital Trends reported that 48% of Americans have shared passwords for TV and movie streaming.† For purposes of this exercise, assume that the 48% figure is correct for the population of adult Americans.

(a)

A random sample of size

n = 500

will be selected from this population and p̂, the proportion who have shared TV and movie streaming passwords, will be calculated. What are the mean and standard deviation of the sampling distribution of p̂? (Round your standard deviation to four decimal places.)

mean

standard deviation

(b)

Is the sampling distribution of p̂ approximately normal for random samples of size

n = 500?

Explain.

The sampling distribution of p̂ is approximately normal because np is less than 10.The sampling distribution of p̂ is approximately normal because np and n(1 − p) are both at least 10.    The sampling distribution of p̂ is not approximately normal because np is less than 10.The sampling distribution of p̂ is not approximately normal because np and n(1 − p) are both at least 10.The sampling distribution of p̂ is not approximately normal because n(1 − p) is less than 10.

(c)

Suppose that the sample size is

n = 125

rather than

n = 500.

What are the values of the mean and standard deviation when

n = 125?

(Round your standard deviation to four decimal places.)

mean

standard deviation

Does the change in sample size affect the mean and standard deviation of the sampling distribution of p̂? If not, explain why not. (Select all that apply.)

When the sample size decreases, the mean increases.

When the sample size decreases, the mean decreases

When the sample size decreases, the mean stays the same. The sampling distribution is always centered at the population mean, regardless of sample size.When the sample size decreases, the standard deviation increases.When the sample size decreases, the standard deviation decreases.When the sample size decreases, the standard deviation stays the same. The standard deviation of the sampling distribution is always the same as the standard deviation of the population distribution, regardless of sample size.

(d)

Is the sampling distribution of p̂ approximately normal for random samples of size

n = 125?

Explain.

The sampling distribution of p̂ is approximately normal because np is less than 10.The sampling distribution of p̂ is approximately normal because np and n(1 − p) are both at least 10.    The sampling distribution of p̂ is not approximately normal because np is less than 10.The sampling distribution of p̂ is not approximately normal because np and n(1 − p) are both at least 10.The sampling distribution of p̂ is not approximately normal because n(1 − p) is less than 10.

Answer 1

(a)

What are the mean and standard deviation of the sampling distribution of p̂?

Mean = 0.48

Standard deviation =

(b)

Is the sampling distribution of p̂ approximately normal for random samples of size n = 500?

Correct option:

The sampling distribution of p̂ is approximately normal because np and n(1 − p) are both at least 10.

(c)

(i)

What are the values of the mean and standard deviation when n = 125?

Mean = 0.48

Standard deviation =

(ii)

Does the change in sample size affect the mean and standard deviation of the sampling distribution of p̂? If not, explain why not. (Select all that apply.)

Correct option:

When the sample size decreases, the mean stays the same. The sampling distribution is always centered at the population mean, regardless of sample size.When the sample size decreases, the standard deviation increases

(d)

Is the sampling distribution of p̂ approximately normal for random samples of size n = 125? Explain.

The sampling distribution of p̂ is approximately normal because np and n(1 − p) are both at least 10.