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Chapter 10: Problem 27

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes \(n\) is the large-sample \(z\) test appropriate: a. \(H_{0}: p=.2, n=25\) b. \(H_{0}: p=.6, n=210\) c. \(H_{0}: p=.9, n=100\) d. \(H_{0}: p=.05, n=75\)

Short Answer

Expert verified
The large-sample z-test can be used for Hypotheses b and c, and it is not appropriate for Hypotheses a and d.

Step by step solution

01

Analyze Hypothesis a

For Hypothesis a, \(H_{0}: p=0.2\) and \(n=25\). Since \(n\) is less than 30, we can't use the large-sample z-test, irrespective of the population proportion \(p\).
02

Analyze Hypothesis b

For Hypothesis b, \(H_{0}: p=0.6\) and \(n=210\). Here, the sample size \(n\) is much greater than 30, and the criterion size \(np=0.6*210=126\) and \(nq=0.4*210=84\) where \(q=1-p\), are both greater than 5. Thus, the large-sample z-test can be used in this scenario.
03

Analyze Hypothesis c

For Hypothesis c, \(H_{0}: p=0.9\) and \(n=100\). Here, although \(n\) is greater than 30, but \(np=0.9*100=90\) and \(nq=0.1*100=10\) where \(q=1-p\), are both greater than 5. This means, we can use the large-sample z-test in this scenario.
04

Analyze Hypothesis d

For Hypothesis d, \(H_{0}: p=0.05\) and \(n=75\). The sample size \(n\) is greater than 30, however the criterion sizes \(np=0.05*75=3.75\) and \(nq=0.95*75=71.25\) where \(q=1-p\), both should be greater than 5, but \(np\) is not. Thus, the large-sample z-test is not appropriate in this scenario.

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Most popular questions from this chapter

The authors of the article "Perceived Risks of Heart Disease and Cancer Among Cigarette Smokers" (journal of the American Medical Association [1999]: 1019-1021) expressed the concern that a majority of smokers do not view themselves as being at increased risk of heart disease or cancer. A study of 737 current smokers selected at random from U.S. households with telephones found that of the 737 smokers surveyed, 295 indicated that they believed they have a higher than average risk of cancer. Do these data suggest that \(p\), the true proportion of smokers who view themselves as being at increased risk of cancer is in fact less than . 5 , as claimed by the authors of the paper? Test the relevant hypotheses using \(\alpha=.05\).

The paper titled "Music for Pain Relief" (The Cochrane Database of Systematic Reviews, April \(19 .\) 2006) concluded, based on a review of 51 studies of the effect of music on pain intensity, that "Listening to music reduces pain intensity levels .... However, the magnitude of these positive effects is small, the clinical relevance of music for pain relief in clinical practice is unclear." Are the authors of this paper claiming that the pain reduction attributable to listening to music is not statistically significant, not practically significant, or neither statistically nor practically significant? Explain.

Medical personnel are required to report suspected cases of child abuse. Because some diseases have symptoms that mimic those of child abuse, doctors who see a child with these symptoms must decide between two competing hypotheses: \(H_{0}:\) symptoms are due to child abuse \(H_{a}:\) symptoms are due to disease (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) The article "Blurred Line Between Illness, Abuse Creates Problem for Authorities" (Macon Telegraph, February 28, 2000) included the following quote from a doctor in Atlanta regarding the consequences of making an incorrect decision: "If it's disease, the worst you have is an angry family. If it is abuse, the other kids (in the family) are in deadly danger." a. For the given hypotheses, describe Type I and Type II errors. b. Based on the quote regarding consequences of the two kinds of error, which type of error does the doctor quoted consider more serious? Explain.

The city council in a large city has become concerned about the trend toward exclusion of renters with children in apartments within the city. The housing coordinator has decided to select a random sample of 125 apartments and determine for each whether children are permitted. Let \(p\) be the proportion of all apartments that prohibit children. If the city council is convinced that \(p\) is greater than \(0.75\), it will consider appropriate legislation. a. If 102 of the 125 sampled apartments exclude renters with children, would a level \(.05\) test lead you to the conclusion that more than \(75 \%\) of all apartments exclude children? b. What is the power of the test when \(p=.8\) and \(\alpha=.05 ?\)

The international polling organization Ipsos reported data from a survey of 2000 randomly selected Canadians who carry debit cards (Canadian Account Habits Survey, July 24, 2006). Participants in this survey were asked what they considered the minimum purchase amount for which it would be acceptable to use a debit card. Suppose that the sample mean and standard deviation were \(\$ 9.15\) and \(\$ 7.60\), respectively. (These values are consistent with a histogram of the sample data that appears in the report.) Do these data provide convincing evidence that the mean minimum purchase amount for which Canadians consider the use of a debit card to be appropriate is less than \(\$ 10\) ? Carry out a hypothesis test with a significance level of \(.01\).
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