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Chapter 8: Problem 23

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

Short Answer

Expert verified
The short answer would be deterministic after manual comparison of the standard deviations calculated in steps 1-3. For example, if the standard deviation obtained from Step 1 is found to be the smallest, the answer would be the combination \(n=40\) and \(p=0.3\).

Step by step solution

01

Compute the standard deviation for the first pair

For \(n=40\) and \(p=0.3\), calculate standard deviation using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\), which gives \(\sigma = \sqrt{\frac{0.3(1-0.3)}{40}}\).
02

Compute the standard deviation for the second pair

For \(n=60\) and \(p=0.4\), again calculate the standard deviation using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\), which gives \(\sigma = \sqrt{\frac{0.4(1-0.4)}{60}}\).
03

Compute the standard deviation for the third pair

For \(n=100\) and \(p=0.5\), calculate standard deviation using the formula \(\sigma = \sqrt{\frac{p(1-p)}{n}}\), which gives \(\sigma = \sqrt{\frac{0.5(1-0.5)}{100}}\).
04

Compare the standard deviations

The sample size and population proportion that gives the smallest standard deviation is the answer. After comparing the three results, select the one with the least standard deviation.

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Most popular questions from this chapter

A random sample of 50 registered voters in a particular city included 32 who favored using city funds for the construction of a new recreational facility. For this sample, \(\hat{p}=\frac{32}{50}=\) 0.64 . If a second random sample of 50 registered voters was selected, would it surprise you if \(\hat{p}\) for that sample was not equal to 0.64 ? Why or why not?

The report "California's Education Skills Gap: Modest Improvements Could Yield Big Gains" (Public Policy Institute of California, April \(16,2008,\) www.ppic.org) states that nationwide, \(61 \%\) of high school graduates go on to attend a two-year or four-year college the year after graduation. The proportion of high school graduates in California who go on to college was estimated to be \(0.55 .\) Suppose that this estimate was based on a random sample of 1,500 California high school graduates. Is it reasonable to conclude that the proportion of California high school graduates who attend college the year after graduation is different from the national figure? (Hint: Use what you know about the sampling distribution of \(\hat{p}\). You might also refer to Example \(8.5 .)\)

The article "Unmarried Couples More Likely to Be Interracial" (San Luis Obispo Tribune, March 13,2002 ) reported that \(7 \%\) of married couples in the United States are mixed racially or ethnically. Consider the population consisting of all married couples in the United States. a. A random sample of \(n=100\) couples will be selected from this population and \(\hat{p},\) the proportion of couples that are mixed racially or ethnically, 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 \(n=100 ?\) Explain. c. Suppose that the sample size is \(n=200\) rather than \(n=\) \(100 .\) 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. d. Is the sampling distribution of \(\hat{p}\) approximately normal for random samples of size \(n=200 ?\) Explain.

A certain chromosome defect occurs in only 1 in 200 adult Caucasian males. A random sample of 100 adult Caucasian males will be selected. The proportion of men in this sample who have the defect, \(\hat{p},\) will be calculated. a. What are the mean and standard deviation of the sampling distribution of \(\hat{p}\) ? b. Is the sampling distribution of \(\hat{p}\) approximately normal? Explain. c. What is the smallest value of \(n\) for which the sampling distribution of \(\hat{p}\) is approximately normal?

If a hurricane were headed your way, would you evacuate? The headline of a press release issued January \(21,2009,\) by the survey research company International Communications Research (icrsurvey.com) states, "Thirty-one Percent of People on High-Risk Coast Will Refuse Evacuation Order, Survey of Hurricane Preparedness Finds." This headline was based on a survey of 5,046 adults who live within 20 miles of the coast in high hurricane risk counties of eight southern states. The sample was selected to be representative of the population of coastal residents in these states, so assume that it is reasonable to regard the sample as if it were a random sample. a. Suppose you are interested in learning about the value of \(p\), the proportion of adults who would refuse to evacuate. This proportion can be estimated using the sample proportion, \(\hat{p} .\) What is the value of \(\hat{p}\) for this sample? b. Based on what you know about the sampling distribution of \(\hat{p}\), is it reasonable to think that the estimate is within 0.03 of the actual value of the population proportion? Explain why or why not.
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