The company Digital Trends reported that 48% of Americans haveshared passwords for TV and movie streaming.†For purposes of thisexercise, assume that the 48% figure is correct for the populationof adult Americans.
(a)
A random sample of size
n = 500
will be selected from this population and p̂, the proportion whohave shared TV and movie streaming passwords, will be calculated.What are the mean and standard deviation of the samplingdistribution of p̂? (Round your standard deviation to four decimalplaces.)
mean
standard deviation
(b)
Is the sampling distribution of p̂ approximately normal forrandom samples of size
n = 500?
Explain.
The sampling distribution of p̂ is approximately normal becausenp is less than 10.The sampling distribution of p̂ isapproximately normal because np and n(1 −p) are both at least 10.    Thesampling distribution of p̂ is not approximately normal becausenp is less than 10.The sampling distribution of p̂ is notapproximately normal because np and n(1 −p) are both at least 10.The sampling distribution of p̂ isnot approximately normal because n(1 − p) is lessthan 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 standarddeviation of the sampling distribution of p̂? If not, explain whynot. (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. Thesampling distribution is always centered at the population mean,regardless of sample size.When the sample size decreases, thestandard deviation increases.When the sample size decreases, thestandard deviation decreases.When the sample size decreases, thestandard deviation stays the same. The standard deviation of thesampling distribution is always the same as the standard deviationof the population distribution, regardless of sample size.
(d)
Is the sampling distribution of p̂ approximately normal forrandom samples of size
n = 125?
Explain.
The sampling distribution of p̂ is approximately normal becausenp is less than 10.The sampling distribution of p̂ isapproximately normal because np and n(1 −p) are both at least 10.    Thesampling distribution of p̂ is not approximately normal becausenp is less than 10.The sampling distribution of p̂ is notapproximately normal because np and n(1 −p) are both at least 10.The sampling distribution of p̂ isnot approximately normal because n(1 − p) is lessthan 10.
