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Chapter 10: Problem 74

The article "Credit Cards and College Students: Who Pays, Who Benefits?" (Journal of College Student Development \([1998]: 50-56\) ) described a study of credit card payment practices of college students. According to the authors of the article, the credit card industry asserts that at most \(50 \%\) of college students carry a credit card balance from month to month. However, the authors of the article report that, in a random sample of 310 college students, 217 carried a balance each month. Does this sample provide sufficient evidence to reject the industry claim?

Short Answer

Expert verified
After calculating, we find that the test statistic value is likely to be significantly greater than the critical value. Thus, based on the sample data, we can reject the null hypothesis that no more than 50% of college students carry a credit card balance. It appears likely that more than 50% of college students carry a credit card balance from month to month.

Step by step solution

01

Setting up the Hypotheses

We set up the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\). In this case, the null hypothesis is that at most 50% of students carry credit card debt, or \(p \leq 0.50\), and the alternative hypothesis is that more than 50% of students carry debt, or \(p > 0.50\).
02

Calculating the Sample Proportion

We calculate the sample proportion \(\hat{p}\) which is equal to the number of 'successes' or students who carry a balance each month divided by the total number of students. In our case, this is \(217/310 \approx 0.6996\).
03

Calculating the Test Statistic

To conduct hypothesis testing, we need to standardize our sample statistic using the formula for the test statistic which is \((\hat{p} - p_0)/\sqrt{ (p_0*(1-p_0)) / n}\), where \(p_0\) is the claimed population proportion, \(\hat{p}\) is the sample proportion and \(n\) is the sample size. Therefore, the test statistic \(z\) is calculated as \((0.6996 - 0.5)/\sqrt{ (0.5*(1-0.5)) / 310}\).
04

Determining the Critical Value

We need to determine the critical Z value for the hypothesis test. Because the problem posits a 'greater than' alternative hypothesis, we would use a one-tailed z-test. This would give us a critical Z value, typically \(\pm 1.645\) for a 5% level of significance in a one-tailed test.
05

Deciding Whether to Reject or Fail to Reject \(H_0\)

We compare the calculated test statistic with the critical value to decide whether to reject or fail to reject the null hypothesis. If the test statistic calculated in Step 3 is greater than the critical value, we reject the null hypothesis. If it is less, we do not reject the null hypothesis.

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Most popular questions from this chapter

According to a survey of 1000 adult Americans conducted by Opinion Research Corporation, 210 of those surveyed said playing the lottery would be the most practical way for them to accumulate \(\$ 200,000\) in net wealth in their lifetime ("One in Five Believe Path to Riches Is the Lottery," San Luis Obispo Tribune, January 11,2006 ). Although the article does not describe how the sample was selected, for purposes of this exercise, assume that the sample can be regarded as a random sample of adult Americans. Is there convincing evidence that more than \(20 \%\) of adult Americans believe that playing the lottery is the best strategy for accumulating \(\$ 200,000\) in net wealth?

Use the definition of the \(P\) -value to explain the following: a. Why \(H_{0}\) would certainly be rejected if \(P\) -value \(=.0003\) b. Why \(H_{0}\) would definitely not be rejected if \(P\) -value \(=\) \(.350\)

Teenagers (age 15 to 20 ) make up \(7 \%\) of the driving population. The article "More States Demand Teens Pass Rigorous Driving Tests" (San Luis Obispo Tribune, January 27,2000 ) described a study of auto accidents conducted by the Insurance Institute for Highway Safety. The Institute found that \(14 \%\) of the accidents studied involved teenage drivers. Suppose that this percentage was based on examining records from 500 randomly selected accidents. Does the study provide convincing evidence that the proportion of accidents involving teenage drivers differs from \(.07\), the proportion of teens in the driving population?

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.

A county commissioner must vote on a resolution that would commit substantial resources to the construction of a sewer in an outlying residential area. Her fiscal decisions have been criticized in the past, so she decides to take a survey of constituents to find out whether they favor spending money for a sewer system. She will vote to appropriate funds only if she can be fairly certain that a majority of the people in her district favor the measure. What hypotheses should she test?
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