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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 100 employees of a large company included 37 who had worked for the company for more than one year. For this sample, \(\hat{p}=\frac{37}{100}=0.37\). If a different random sample of 100 employees were selected, would you expect that \(\hat{p}\) for that sample would also be \(0.37 ?\) Explain why or why not.

In a study of pet owners, it was reported that \(24 \%\) celebrate their pet's birthday (Pet Statistics, Bissell Homecare, Inc., 2010 ). Suppose that this estimate was from a random sample of 200 pet owners. Is it reasonable to conclude that the proportion of all pet owners who celebrate their pet's birthday is less than \(0.25 ?\) Use what you know about the sampling distribution of \(\hat{p}\) to support your answer.

A random sample is to be selected from a population that has a proportion of successes \(p=0.65\). Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following sample sizes: a. \(n=10\) d. \(n=50\) b. \(n=20\) e. \(n=100\) c. \(n=30\) f. \(n=200\)

In a national survey of 2,013 American adults, 1,283 indicated that they believe that rudeness is a more serious problem than in past years (Associated Press, April 3,2002 ). Assume that it is reasonable to regard this sample as a random sample of adult Americans. Is it reasonable to conclude that the proportion of adults who believe that rudeness is a worsening problem is greater than \(0.5 ?\) (Hint: Use what you know about the sampling distribution of \(\hat{p} .\) You might also refer to Example 8.5.)

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?
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