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Chapter 10: Problem 42

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?

Short Answer

Expert verified
The answer depends on the computed p-value. If it's smaller than 0.05, it provides convincing evidence that the proportion of accidents involving teenage drivers significantly differs from the proportion of teenagers in the driving population (0.07). If it's larger, then it doesn't provide convincing evidence.

Step by step solution

01

Formulate the Hypotheses

The null hypothesis (\(H_0\)) is that the proportion of accidents involving teenagers is the same as the proportion of teenagers in the population, i.e. 0.07. The alternative hypothesis (\(H_a\)) is that the proportion of accidents involving teenagers is different from 0.07, hence the two tailed test. \nSo, \(H_0: P = 0.07\) and \(H_a: P \neq 0.07\).
02

Calculate the Test Statistic

The test statistic z can be calculated using the formula:\[z = \frac{(P - P_0) \sqrt {n}}{\sqrt{P_0(1 - P_0)}}\]Where in this case, \(P = 0.14\) (from the sample), \(P_0 = 0.07\) (from the population), and \(n = 500\) (sample size).
03

Compute the p-value

The p-value is found by looking up the absolute value of test statistic z in the standard normal (z) table or calculator. Remember it鈥檚 a two-tailed test, so the p-value is two times the value found on the table.
04

Reject or Fail to Reject the Null Hypothesis

If the p-value is smaller than the significance level (\( \alpha \)), in general, a 0.05 level is used, we reject the null hypothesis. If it鈥檚 larger, we fail to reject the null hypothesis.

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Most popular questions from this chapter

An automobile manufacturer is considering using robots for part of its assembly process. Converting to robots is an expensive process, so it will be undertaken only if there is strong evidence that the proportion of defective installations is lower for the robots than for human assemblers. Let \(\pi\) denote the true proportion of defective installations for the robots. It is known that human assemblers have a defect proportion of 02 . a. Which of the following pairs of hypotheses should the manufacturer test: $$ H_{0}: \pi=.02 \text { versus } H_{s}: \pi<.02 $$ or $$ H_{0}: \pi=.02 \text { versus } H_{a}: \pi>.02 $$ Explain your answer. b. In the context of this exercise, describe Type I and Type II errors. c. Would you prefer a test with \(\alpha=.01\) or \(\alpha=.1 ?\) Explain your reasoning.

Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most \(150^{\circ} \mathrm{F}\), there will be no negative effects on the river's ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above \(150^{\circ} \mathrm{F}\), researchers will take 50 water samples at randomly selected times and record the temperature of each sample. The resulting data will be used to test the hypotheses \(H_{0}: \mu=150^{\circ} \mathrm{F}\) versus \(H_{a}: \mu>150^{\circ} \mathrm{F}\). In the context of this example, describe Type I and Type II errors. Which type of error would you consider more serious? Explain.

When a published article reports the results of many hypothesis tests, the \(P\) -values are not usually given. Instead, the following type of coding scheme is frequently used: \({ }^{*} p=.05,{ }^{* *} p=.01,{ }^{* * *} p=.001,{ }^{* * * *} p=.0001\). Which of the symbols would be used to code for each of the following \(P\) -values? a. \(.037\) c. 072 b. \(.0026\) d. \(.0003\)

Consider the following quote from the article "Review Finds No Link Between Vaccine and Autism" (San Luis Obispo Tribune, October 19,2005 ): " 'We found no evidence that giving MMR causes Crohn's disease and/or autism in the children that get the MMR,' said Tom Jefferson, one of the authors of The Cochrane Review. 'That does not mean it doesn't cause it. It means we could find no evidence of it." (MMR is a measles-mumps-rubella vaccine.) In the context of a hypothesis test with the null hypothesis being that MMR does not cause autism, explain why the author could not just conclude that the MMR vaccine does not cause autism.

The state of Georgia's HOPE scholarship program guarantees fully paid tuition to Georgia public universities for Georgia high school seniors who have a B average in academic requirements as long as they maintain a B average in college. Of 137 randomly selected students enrolling in the Ivan Allen College at the Georgia Institute of Technology (social science and humanities majors) in 1996 who had a B average going into college, \(53.2 \%\) had a GPA below \(3.0\) at the end of their first year ("Who Loses HOPE? Attrition from Georgia's College Scholarship Program," Southern Economic Journal [1999]: 379-390). Do these data provide convincing evidence that a majority of students at Ivan Allen College who enroll with a HOPE scholarship lose their scholarship?
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