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Chapter 10: Problem 58

Exercises 58 to 60 refer to the following setting. A study of road rage asked random samples of \(596 \mathrm{men}\) and 523 women about their behavior while driving. Based on their answers, each person was assigned a road rage score on a scale of 0 to \(20 .\) The participants were chosen by random digit dialing of phone numbers. The researchers performed a test of the following hypotheses: \(H_{0}: \mu_{M}=\mu_{F}\) versus \(H_{a^{*}} \mu_{M} \neq \mu_{F}\) Which of the following describes a Type II error in the context of this study? (a) Finding convincing evidence that the true means are different for males and females, when in reality the true means are the same (b) Finding convincing evidence that the true means are different for males and females, when in reality the true means are different (c) Not finding convincing evidence that the true means are different for males and females, when in reality the true means are the same (d) Not finding convincing evidence that the true means are different for males and females, when in reality the true means are different (e) Not finding convincing evidence that the true means are different for males and females, when in reality there is convincing evidence that the true means are different

Short Answer

Expert verified
(d) Not finding convincing evidence that the true means are different when in reality they are.

Step by step solution

01

Understanding Type II Error

A Type II error occurs when the null hypothesis is not rejected, even though the alternative hypothesis is true. In statistical terms, it implies we fail to recognize a difference that actually exists.
02

Identifying Null and Alternative Hypotheses

In this study, the null hypothesis \( H_0 \) states that the true means of road rage scores for men and women are equal, \( \mu_M = \mu_F \). The alternative hypothesis \( H_a \) suggests that the true means are not equal, \( \mu_M eq \mu_F \).
03

Linking Context to Type II Error

Relating the definition of a Type II error to this specific study, we would not find convincing evidence that \( \mu_M eq \mu_F \) (fail to reject \( H_0 \)), when, in reality, \( \mu_M eq \mu_F \) (\( H_a \) is true).
04

Analyzing the Options

Reviewing the options provided: - (a) Suggests rejecting \( H_0 \) when it's true.- (b) Suggests rejecting \( H_0 \) when it's false.- (c) Suggests failing to reject \( H_0 \) when it's true.- (d) Suggests failing to reject \( H_0 \) when \( H_a \) is true.- (e) Suggests failing to reject \( H_0 \) when there is evidence against it.Option (d) aligns with a Type II error as it describes failing to recognize a true difference.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing
Hypothesis testing is a critical process in statistical analysis that helps us make informed decisions based on data samples. The main goal of hypothesis testing is to determine whether there is enough evidence in a sample to infer that a certain condition is true for the entire population.

The process generally involves the following steps:
  • Formulating a null and an alternative hypothesis.
  • Collecting data and selecting an appropriate statistical test.
  • Calculating a test statistic and corresponding p-value.
  • Deciding whether to reject or fail to reject the null hypothesis based on the p-value and chosen significance level.
Using hypothesis testing, researchers can objectively assess evidence and make conclusions about population parameters, which is particularly valuable in fields like medicine, psychology, and social sciences.
Null Hypothesis
The null hypothesis, denoted as \( H_0 \), is a statement suggesting that there is no effect or no difference between groups or variables being measured. It serves as a default or starting assumption in hypothesis testing. For example, in the context of the road rage study, the null hypothesis is \( \mu_M = \mu_F \), meaning that the average road rage scores for men and women are the same.

Testing the null hypothesis involves determining whether there is sufficient statistical evidence to reject it. If researchers can show that the observed data is highly unlikely under the assumption of the null hypothesis, they may reject it in favor of the alternative hypothesis. However, if the data does not provide convincing evidence against the null hypothesis, it is not rejected, which does not necessarily prove it true, but rather suggests the data does not sufficiently indicate a difference.
Alternative Hypothesis
The alternative hypothesis, represented as \( H_a \) or \( H_1 \), contradicts the null hypothesis. It implies that there is a statistically significant effect or difference between the groups or variables studied. In the road rage study, the alternative hypothesis is \( \mu_M eq \, \mu_F \), suggesting that men and women have different average road rage scores.

This hypothesis is what the researcher aims to support; hence, statistical tests are arranged to find evidence favoring the alternative hypothesis. When the null hypothesis is rejected based on the test results, it indicates sufficient evidence to accept the alternative hypothesis, adding value to scientific knowledge and allowing researchers to draw meaningful conclusions from their data.
Road Rage Study
The road rage study serves as a practical example of hypothesis testing in real-world research. This study involved randomly selecting participants through phone number digit dialing, intending to eliminate bias and create a representative sample. Subjects were assigned a road rage score on a scale from 0 to 20 based on specific criteria.

In this study, researchers wanted to investigate whether there are significant differences in road rage behaviors between men and women. By assessing the mean scores for both genders, they tested their hypotheses to determine any statistically significant disparity. The Type II error description in this context is particularly insightful, as it directly connects to the possibility of failing to detect a true difference in road rage scores when one exists, emphasizing the importance of understanding error types in hypothesis testing to correctly interpret study results.

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Most popular questions from this chapter

There are two common methods for measuring the concentration of a pollutant in fish tissue. Do the two methods differ, on average? You apply both methods to each fish in a random sample of 18 carp and use (a) the paired \(t\) test for \(\mu_{d}\) (b) the one-sample \(z\) test for \(p\). (c) the two-sample \(t\) test for \(\mu_{1}-\mu_{2}\). (d) the two-sample \(z\) test for \(p_{1}-p_{2}\). (e) none of these.

State which inference procedure from Chapter \(8,9,\) or 10 you would use. Be specific. For example, you might say, "Two-sample z test for the difference between two proportions." You do not need to carry out any procedures. Which inference method? (a) How do young adults look back on adolescent romance? Investigators interviewed 40 couples in their midtwenties. The female and male partners were interviewed separately. Each was asked about his or her current relationship and also about a romantic relationship that lasted at least two months when they were aged 15 or \(16 .\) One response variable was a measure on a numerical scale of how much the attractiveness of the adolescent partner mattered. You want to find out how much men and women differ on this measure. (b) Are more than \(75 \%\) of Toyota owners generally satisfied with their vehicles? Let's design a study to find out. We'll select a random sample of 400 Toyota owners. Then we'll ask each individual in the sample: "Would you say that you are generally satisfied with your Toyota vehicle?" (c) Are male college students more likely to binge drink than female college students? The Harvard School of Public Health surveys random samples of male and female undergraduates at four-year colleges and universities about whether they have engaged in binge drinking. (d) A bank wants to know which of two incentive plans will most increase the use of its credit cards and by how much. It offers each incentive to a group of current credit card customers, determined at random, and compares the amount charged during the following six months.

Exercises 58 to 60 refer to the following setting. A study of road rage asked random samples of \(596 \mathrm{men}\) and 523 women about their behavior while driving. Based on their answers, each person was assigned a road rage score on a scale of 0 to \(20 .\) The participants were chosen by random digit dialing of phone numbers. The researchers performed a test of the following hypotheses: \(H_{0}: \mu_{M}=\mu_{F}\) versus \(H_{a^{*}} \mu_{M} \neq \mu_{F}\) The \(P\) -value for the stated hypotheses is 0.002 . Interpret this value in the context of this study. (a) Assuming that the true mean road rage score is the same for males and females, there is a 0.002 probability of getting a difference in sample means. (b) Assuming that the true mean road rage score is the same for males and females, there is a 0.002 probability of getting an observed difference at least as extreme as the observed difference. (c) Assuming that the true mean road rage score is different for males and females, there is a 0.002 probability of getting an observed difference at least as extreme as the observed difference. (d) Assuming that the true mean road rage score is the same for males and females, there is a 0.002 probability that the null hypothesis is true. (e) Assuming that the true mean road rage score is the same for males and females, there is a 0.002 probability that the alternative hypothesis is true.

Broken crackers We don't like to find broken crackers when we open the package. How can makers reduce breaking? One idea is to microwave the crackers for 30 seconds right after baking them. Breaks start as hairline cracks called "checking." Randomly assign 65 newly baked crackers to the microwave and another 65 to a control group that is not microwaved. After one day, none of the microwave group and 16 of the control group show checking.

Paying for college College financial aid offices expect students to use summer eamings to help pay for college. But how large are these eamings? One large university studied this question by asking a random sample of 1296 students who had summer jobs how much they earned. The financial aid office separated the responses into two groups based on gender. Here are the data in summary form: \({ }^{30}\) $$ \begin{array}{lccc} \hline \text { Group } & n & \bar{\chi} & s_{\chi} \\ \text { Males } & 675 & \$ 1884.52 & \$ 1368.37 \\ \text { Females } & 621 & \$ 1360.39 & \$ 1037.46 \\ \hline \end{array} $$ (a) How can you tell from the summary statistics that the distribution of earnings in each group is strongly skewed to the right? The use of two-sample \(t\) procedures is still justified. Why? (b) Construct and interpret a \(90 \%\) confidence interval for the difference between the mean summer earnings of male and female students at this university. (c) Interpret the \(90 \%\) confidence level in the context of this study.
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