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Chapter 10: Problem 89

Past experience is that when individuals are approached with a request to fill out and return a particular questionnaire in a provided stamped and addressed envelope, the response rate is \(40 \%\). An investigator believes that if the person distributing the questionnaire were stigmatized in some obvious way, potential respondents would feel sorry for the distributor and thus tend to respond at a rate higher than \(40 \%\). To test this theory, a distributor wore an eye patch. Of the 200 questionnaires distributed by this individual, 109 were returned. Does this provide evidence that the response rate in this situation is greater than the previous rate of \(40 \%\) ? State and test the appropriate hypotheses using a significance plevel of 0.05 .

Short Answer

Expert verified
The result of the test provides strong evidence to reject the null hypothesis and conclude that the response rate is indeed higher when the distributor wears an eye patch, at a significance level of 0.05.

Step by step solution

01

Set Up the Hypotheses

The null hypothesis (H0) states that the response rate does not change, i.e., the response rate is equal to 40%. The alternative hypothesis (H1) states that the response rate is higher than 40% with an eye patch.\[ H0: p = 0.4 \]\[ H1: p > 0.4 \]
02

Calculate the Test Statistic

The test statistic for a hypothesis test about a proportion is a z-score (z). \[ z = \frac{(p_1 - p_o) }{\sqrt{\frac{(p_o(1 - p_o))}{n}}\] which equals to \[ z = \frac{(0.545 - 0.4) }{\sqrt{\frac{(0.4(1 - 0.4))}{200}}\] After calculating, the z-score is approximately 4.36.
03

Determine the P-value

The P-value is the probability that a z-score is more than the absolute value of the test statistic, considering the null hypothesis is true. For this two-sided test, the P-value is the area to the right of our test statistic (z = 4.36). Using standard statistical software or Z-tables, the P-value is approximately 0.00001.
04

Make the Decision

Compare the P-value to the significance level and make the decision about the null hypothesis. The P-value is less than the significance level of 0.05, so the decision is to reject the null hypothesis.
05

Draw a Conclusion

Given the result of the test, there is enough evidence to support the claim that the response rate is greater than 40% when the distributor wears an eye patch.

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Most popular questions from this chapter

The paper "College Students' Social Networking Experiences on Facebook" (Journal of Applied Developmental Psychology [2009]: 227-238) summarized a study in which 92 students at a private university were asked how much time they spent on Facebook on a typical weekday. The researchers were interested in estimating the average time spent on Facebook by students at this university.

The paper "Teens and Distracted Driving" (Pew Internet \& American Life Project, 2009) reported that in a representative sample of 283 American teens ages 16 to \(17,\) there were 74 who indicated that they had sent a text message while driving. For purposes of this exercise, assume that this sample is a random sample of 16 - to 17 -year-old Americans. Do these data provide convincing evidence that more than a quarter of Americans ages 16 to 17 have sent a text message while driving? Test the appropriate hypotheses using a significance level of 0.01 . (Hint: See Example 10.11 )

For which of the following \(P\) -values will the null hypothesis be rejected when performing a test with a significance level of \(0.05 ?\) a. 0.001 d. 0.047 b. 0.021 e. 0.148 c. 0.078

The article "Cops Get Screened for Digital Dirt" (USA Today, Nov. 12,2010 ) summarizes a report on law enforcement agency use of social media to screen applicants for employment. The report was based on a survey of 728 law enforcement agencies. One question on the survey asked if the agency routinely reviewed applicants' social media activity during background checks. For purposes of this exercise, suppose that the 728 agencies were selected at random and that you want to use the survey data to decide if there is convincing evidence that more than \(25 \%\) of law enforcement agencies review applicants' social media activity as part of routine background checks. a. Describe the shape, center, and spread of the sampling distribution of \(\hat{p}\) for samples of size 728 if the null hypothesis \(H_{0}: p=0.25\) is true. b. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.27\) for a sample of size 728 if the null hypothesis \(H_{0}: p=0.25\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.31\) for a sample of size 728 if the null hypothesis \(H_{0}: p=0.25\) were true? Explain why or why not. d. The actual sample proportion observed in the study was \(\hat{p}=0.33 .\) Based on this sample proportion, is there convincing evidence that more than \(25 \%\) of law enforcement agencies review social media activity as part of background checks, or is this sample proportion consistent with what you would expect to see when the null hypothesis is true?

Duck hunting in populated areas faces opposition on the basis of safety and environmental issues. In a survey to assess public opinion regarding duck hunting on Morro Bay (located along the central coast of California), a random sample of 750 local residents included 560 who strongly opposed hunting on the bay. Does this sample provide convincing evidence that a majority of local residents oppose hunting on Morro Bay? Test the relevant hypotheses using \(\alpha=0.01\).
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