The report "New Study Shows Need for Americans to Focus on Securing Online
Accounts and Backing Up Critical Data" (PRNewswire, October 29,2009 ) reported
that only \(25 \%\) of Americans change computer passwords quarterly, in spite
of a recommendation from the National Cyber Security
Alliance that passwords be changed at least once every 90 days. For purposes
of this exercise, assume that the \(25 \%\) figure is correct for the population
of adult Americans.
a. A random sample of size \(n=200\) will be selected from this population and
\(\hat{p}\), the proportion who change passwords quarterly, will be calculated.
What are the mean and standard deviation of the sampling distribution of
\(\hat{p} ?\)
b. Is the sampling distribution of \(\hat{p}\) approximately normal for random
samples of size \(n=200 ?\) Explain.
c. Suppose that the sample size is \(n=50\) rather than \(n=200 .\) Does the
change in sample size affect the mean and standard deviation of the sampling
distribution of \(\hat{p} ?\) If so, what are the new values of the mean and
standard deviation? If not, explain why not.
d. Is the sampling distribution of \(\hat{p}\) approximately normal for random
samples of size \(n=50 ?\) Explain.
