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A random sample of 1000 students at a large college included 428 who had one or more credit cards. For this sample, \(\hat{p}=\frac{428}{1000}=0.428 .\) If another random sample of 1000 students from this university were selected, would you expect that \(\hat{p}\) for that sample would also be 0.428 ? Explain why or why not.

Consider the following statement: The Department of Motor Vehicles reports that the proportion of all vehicles registered in California that are imports is \(0.22 .\) a. Is the number that appears in boldface in this statement a sample proportion or a population proportion? b. Which of the following use of notation is correct, \(p=0.22\) or \(\hat{p}=0.22 ?\) (Hint: See definitions and notation on page \(403 .\) )

Consider the two relative frequency histograms at the top of the next page. The histogram on the left was constructed by selecting 100 different random samples of size 40 from a population consisting of \(20 \%\) part-time students and \(80 \%\) full-time students. For each sample, the sample proportion of part-time students, \(\hat{p},\) was calculated. The \(100 \hat{p}\) values were used to construct the histogram. The histogram on the right was constructed in a similar way, but using samples of size 70 . a. Which of the two histograms indicates that the value of \(\hat{p}\) has smaller sample-to-sample variability? How can you tell? b. For which of the two sample sizes, \(n=40\) or \(n=70,\) do you think the value of \(\hat{p}\) would be less likely to be close to \(0.20 ?\) What about the given histograms supports your choice?

Explain what the term sampling variability means in the context of using a sample proportion to estimate a population proportion.

Explain what it means when we say the value of a sample statistic varies from sample to sample.

Consider the following statement: In a sample of 20 passengers selected from those who flew from Dallas to New York City in April \(2017,\) the proportion who checked luggage was \(\mathbf{0} . \mathbf{4 5}\). a. Is the number that appears in boldface in this statement a sample proportion or a population proportion? b. Which of the following use of notation is correct, \(p=0.45\) or \(\hat{p}=0.45 ?\)

The U.S. Census Bureau reported that in 2015 the proportion of adult Americans age 25 and older who have a bachelor's degree or higher is 0.325 ("Educational Attainment in the United States: 2015," www.census.gov, retrieved January 22,2017 ). Consider the population of all adult Americans age 25 and over in 2015 and define \(\hat{p}\) to the proportion of people in a random sample from this population who have a bachelor's degree or higher. a. Would \(\hat{p}\) based on a random sample of only 10 people from this population have a sampling distribution that is approximately normal? Explain why or why not. b. What are the mean and standard deviation of the sampling distribution of \(\hat{p}\) if the sample size is \(400 ?\) c. Suppose that the sample size is \(n=200\) rather than \(n=\) \(400 .\) 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 for the mean and standard deviation? If not, explain why not.

For which of the following combinations of sample size and population proportion would the standard deviation of \(\hat{p}\) be smallest? $$ \begin{array}{ll} n=40 & p=0.3 \\ n=60 & p=0.4 \\ n=100 & p=0.5 \end{array} $$

Explain why the standard deviation of \(\hat{p}\) is equal to 0 when the population proportion is equal to 1 .

For which of the following sample sizes would the sampling distribution of \(\hat{p}\) be approximately normal when $$ \begin{array}{c} p=0.2 ? \text { When } p=0.8 ? \text { When } p=0.6 ? \\ n=10 \quad n=25 \\ n=50 \quad n=100 \end{array} $$