91Ó°ÊÓ

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.

Step by step solution

01

Understanding the Hypothesis Test

We are testing the null hypothesis \(H_{0}: \mu_{M} = \mu_{F}\) against the alternative hypothesis \(H_{a}: \mu_{M} eq \mu_{F}\), where \(\mu_M\) is the mean road rage score for men and \(\mu_F\) is the mean road rage score for women. The null hypothesis asserts there is no difference between the mean scores, while the alternative suggests a difference exists.

02

Understanding the P-Value

The \(P\)-value, given as 0.002, quantifies the probability of observing a test statistic as extreme as, or more extreme than, the one observed in the sample, under the assumption that the null hypothesis is true. A small \(P\)-value indicates that the observed data is unlikely under the null hypothesis.

03

Identify the Correct Interpretation

Out of the given options, the interpretation closest to the definition of a \(P\)-value is option (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." This aligns with the concept of a \(P\)-value testing the hypothesis assuming \(H_0\) is true.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

P-value

The P-value in hypothesis testing is a crucial concept to understand. It tells us how likely the observed data would occur if the null hypothesis were true. In simpler terms, it measures how well the sample data supports the null hypothesis. For our study, the P-value is given as 0.002. This low P-value suggests that the observed differences in road rage scores between men and women are unlikely if we assume the null hypothesis to be correct.
When interpreting a P-value, remember that:

• A small P-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so we reject the null hypothesis.
• A large P-value (> 0.05) suggests weak evidence against the null hypothesis, so we fail to reject it.
• The threshold (like 0.05) is called the significance level, and it is chosen based on how much uncertainty we are willing to accept in our conclusions.

Therefore, the 0.002 P-value in this study strongly suggests a meaningful difference in road rage scores between the genders, prompting researchers to likely reject the null hypothesis.

Null Hypothesis

The null hypothesis (often denoted as \(H_0\)) is the starting assumption for statistical tests. It suggests no effect or no difference exists between groups or conditions being compared. In the road rage study, the null hypothesis \(H_0: \mu_{M} = \mu_{F}\) claims that the average road rage scores are the same for men and women.
The null hypothesis serves several purposes:

• It acts as a baseline that we test against using the sample data.
• It provides a specific scenario under which we calculate the P-value.
• In many tests, it is favored or assumed true unless evidence strongly suggests otherwise.

Rejecting the null hypothesis does not prove the null hypothesis false; it just suggests that the observed data are not consistent with the assumption of no difference.

Alternative Hypothesis

The alternative hypothesis (denoted as \(H_a\)) is what researchers typically want to prove. It suggests that there is an effect or a difference. In our example, the alternative hypothesis \(H_a: \mu_M eq \mu_F\) asserts that there is a difference in mean road rage scores between men and women.
Here are key points about the alternative hypothesis:

• It is what we accept when the null hypothesis is rejected based on the P-value and significance level.
• It provides the basis for concluding that the observed effect is statistically significant.
• Unlike the null hypothesis, the alternative hypothesis is usually not a statement of no effect.

In this exercise, the extremely low P-value of 0.002 provides enough evidence for the researchers to consider accepting the alternative hypothesis, indicating a difference in road rage scores based on gender.