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Chapter 9: Problem 4

A researcher wants to estimate the proportion of students enrolled at a university who eat fast food more than three times in a typical week. Would the standard error of the sample proportion \(\hat{p}\) be smaller for random samples of size \(n=50\) or random samples of size \(n=200 ?\)

Short Answer

Expert verified
The standard error of the sample proportion, \(\hat{p}\), would be smaller for random samples of size \(n=200\) compared to random samples of size \(n=50\).

Step by step solution

01

Understand the Standard Error of the Sample Proportion

The standard error of the sample proportion (\(\hat{p}\)) measures the variability or dispersion of the sampling distribution. It is given by the formula \(\sqrt{(\hat{p}(1-\hat{p})/n)}\), where \(\hat{p}\) is the sample proportion and n is the sample size.
02

Effect of Sample Size on Standard Error

In the formula for the standard error, n (the sample size) is in the denominator. This means that as n increases, the total value of the standard error decreases. That is, a larger sample size results in a smaller standard error.
03

Compare the Standard Errors for Different Sample Sizes

If we compare a sample size of 50 to a sample size of 200, since 200 is larger than 50, the standard error of the sample proportion will be smaller for a sample size of 200 compared to a sample size of 50.

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Most popular questions from this chapter

In a study of 1,710 schoolchildren in Australia (Herald Sun, October 27,1994 ), 1,060 children indicated that they normally watch TV before school in the morning. (Interestingly, only \(35 \%\) of the parents said their children watched TV before school.) Construct and interpret a \(95 \%\) confidence interval for the proportion of all Australian children who say they watch TV before school. In order for the method used to construct the interval to be valid, what assumption about the sample must be reasonable?

Consider taking a random sample from a population with \(p=0.70\). a. What is the standard error of \(\hat{p}\) for random samples of size \(100 ?\) b. Would the standard error of \(\hat{p}\) be smaller for samples of size 100 or samples of size \(400 ?\) c. Does decreasing the sample size by a factor of \(4,\) from 400 to 100 , result in a standard error of \(\hat{p}\) that is four times as large?

A random sample will be selected from the population of all students enrolled at a large college. The sample proportion \(\hat{p}\) will be used to estimate \(p,\) the proportion of all students who use public transportation to travel to campus. For which of the following situations will the estimate tend to be closest to the actual value of \(p ?\) i. \(\quad n=300, p=0.3\) ii. \(\quad n=700, p=0.2\) iii. \(n=1,000, p=0.1\)

The article "Hospitals Dispute Medtronic Data on Wires" (The Wall Street Journal, February 4, 2010) describes several studies of the failure rate of defibrillators used in the treatment of heart problems. In one study conducted by the Mayo Clinic, it was reported that failures within the first 2 years were experienced by 18 of 89 patients under 50 years old and 13 of 362 patients age 50 and older. Assume that these two samples are representative of patients who receive this type of defibrillator in the two age groups. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of patients under 50 years old who experience a failure within the first 2 years. b. Construct and interpret a \(99 \%\) confidence interval for the proportion of patients age 50 and older who experience a failure within the first 2 years. c. Suppose that the researchers wanted to estimate the proportion of patients under 50 years old who experience this type of failure with a margin of error of \(0.03 .\) How large a sample should be used? Use the given study results to obtain a preliminary estimate of the population proportion.

In an AP-AOL sports poll (Associated Press, December 18,2005\(), 394\) of 1,000 randomly selected U.S. adults indicated that they considered themselves to be baseball fans. Of the 394 baseball fans, 272 stated that they thought the designated hitter rule should either be expanded to both baseball leagues or eliminated. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of U.S. adults who consider themselves to be baseball fans. b. Construct and interpret a \(95 \%\) confidence interval for the proportion of baseball fans who think the designated hitter rule should be expanded to both leagues or eliminated. c. Explain why the confidence intervals of Parts (a) and (b) are not the same width even though they both have a confidence level of \(95 \%\).
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