91Ó°ÊÓ

The paper "The Observed Effects of Teenage Passengers on the Risky Driving Behavior of Teenage Drivers" (Accident Analysis and Prevention [2005]: 973-982) investigated the driving behavior of teenagers by observing their vehicles as they left a high school parking lot and then again at a site approximately \(\frac{1}{2}\) mile from the school. Assume that it is reasonable to regard the teen drivers in this study as representative of the population of teen drivers. Use a .01 level of significance for any hypothesis tests. a. Data consistent with summary quantities appearing in the paper are given in the accompanying table. The measurements represent the difference between the observed vehicle speed and the posted speed limit (in miles per hour) for a sample of male teenage drivers and a sample of female teenage drivers. Do these data provide convincing support for the claim that, on average, male teenage drivers exceed the speed limit by more than do female teenage drivers? $$ \begin{array}{cc} \hline \text { Male Driver } & \text { Female Driver } \\ \hline 1.3 & -0.2 \\ 1.3 & 0.5 \\ 0.9 & 1.1 \\ 2.1 & 0.7 \\ 0.7 & 1.1 \\ 1.3 & 1.2 \\ 3 & 0.1 \\ 1.3 & 0.9 \\ 0.6 & 0.5 \\ 2.1 & 0.5 \\ \hline \end{array} $$ b. Consider the average miles per hour over the speed limit for teenage drivers with passengers shown in the table at the top of the following page. For purposes of this exercise, suppose that each driver-passenger combination mean is based on a sample of size \(n=40\) and that all sample standard deviations are equal to .8 . $$ \begin{array}{l|cc} & \text { Male Passenger } & \text { Female Passenger } \\ \hline \text { Male Driver } & 5.2 & .3 \\ \text { Female Driver } & 2.3 & .6 \\ \hline \end{array} $$ i. Is there sufficient evidence to conclude that the average number of miles per hour over the speed limit is greater for male drivers with male passengers than it is for male drivers with female passengers? ii. Is there sufficient evidence to conclude that the average number of miles per hour over the speed limit is greater for female drivers with male passengers than it is for female drivers with female passengers? iii. Is there sufficient evidence to conclude that the average number of miles per hour over the speed limit is smaller for male drivers with female passengers than it is for female drivers with male passengers? c. Write a few sentences commenting on the effects of gender on teenagers driving with passengers.

Step by step solution

01

Analyze Data For Male and Female Teenage Drivers

The data given is a set of speed limit excesses by male and female teenage drivers. You first need to compute the sample means, \(\overline{x}_M\) and \(\overline{x}_F\), and the sample standard deviations, \(s_M\) and \(s_F\), for male and female drivers, respectively.

02

Perform Two-Sample t-Test

You conduct a two-sample t-test to compare the means. The NULL hypothesis, H0, states that the average speed limit excess of male drivers equals to that of female drivers. The ALTERNATIVE hypothesis, Ha, states that the average speed limit excess of male drivers is greater than that of female drivers. You use a .01 level of significance. After calculating the test statistic, you compare it with the critical value from t-distribution table. If the test statistic > critical value, reject H0.

03

Analyze Data For Teenage Drivers with Passengers

The given data consists of the average speed limit excess by male and female drivers with male and female passengers. It's also given that each mean is based on a sample of size \(n=40\) and that all sample standard deviations equals to .8.

04

Perform Paired t-Tests

Conduct three separate paired t-tests: (i) for male drivers with male passengers vs male drivers with female passengers, (ii) for female drivers with male passengers vs female drivers with female passengers, and (iii) for male drivers with female passengers vs female drivers with male passengers. You follow the same procedure as in step 2, formulating the null and alternative hypotheses for each test, calculating the test statistics and comparing them to the critical values. You reject the null hypothesis if the calculated test statistic > critical value.

05

Comment on the Effects of Gender on Teenagers Driving with Passengers

Based on the results of your hypothesis tests, you make inferences about the effect of the gender of the driver and the passenger on the amount by which the speed limit is exceeded.

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Key Concepts

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

Two-Sample t-Test

A Two-Sample t-Test is a statistical method used to determine if there are significant differences between the means of two independent groups. In the context of teenage driving behavior, we compare the average speed excess of male and female drivers. The goal is to assess if male teenagers, on average, drive faster than females.

To perform this test, we follow these general steps:

• Calculate the mean and standard deviation for both groups.
• Formulate hypotheses: The null hypothesis (H_0) states there is no difference in means, while the alternative hypothesis (H_a) suggests the male mean is greater.
• Determine the t-statistic using the formula:\[t = \frac{\overline{x}_M - \overline{x}_F}{\sqrt{ \frac{s_M^2}{n_M} + \frac{s_F^2}{n_F} }}\]where \overline{x}_M and \overline{x}_F are the sample means, and s_M and s_F are the standard deviations for males and females respectively.
• Compare the t-statistic to a critical value from the t-distribution table at a .01 significance level.

If the t-statistic is greater than the critical value, we reject H_0, providing evidence that male teenagers exceed speed limits more than females.

Paired t-Test

A Paired t-Test is used when comparing two related groups. In this exercise, it's applied to understand teenage speeding behavior with different passenger genders. The test helps us determine if passengers affect driving speeds differently based on gender.

Here's a breakdown of the paired t-test process:

• Identify paired samples, such as male drivers with both male and female passengers.
• Compute the difference in their speed excess for each pair.
• Set hypotheses: The null hypothesis (H_0) implies there's no difference between groups, while the alternative hypothesis (H_a) indicates a difference exists.
• Calculate the t-statistic:\[t = \frac{\overline{d}}{\frac{s_d}{\sqrt{n}}}\]where \overline{d} is the average difference in pairs and \s_d is the standard deviation of these differences.
• Compare the test statistic to the critical value at .01 significance level.

If the test statistic surpasses the critical value, we reject H_0, indicating that passenger gender significantly influences speeding.

Statistical Analysis

Statistical Analysis involves using mathematical techniques to interpret data, allowing us to make informed decisions based on evidence. In the context of teenage driving, statistical tests like the Two-Sample t-Test and Paired t-Test help us discern patterns and make predictions.

Here's why statistical analysis is critical in studies like these:

• It ensures objectivity, letting us test hypotheses about driving behavior systematically.
• Calculating statistics such as mean and standard deviation offers a snapshot of data variability.
• Performing hypothesis tests guides conclusions on whether observed differences are significant or just due to random chance.

Ultimately, statistical analysis transforms raw observations into meaningful insights about how teenagers' driving is influenced by factors such as gender and the presence of passengers.

Teenage Driving Behavior

Understanding Teenage Driving Behavior is essential for improving road safety. Young drivers often exhibit distinct patterns and tendencies, such as exceeding speed limits, which are influenced by factors like gender and peer presence.

Research indicates:

• Male teenagers typically display a higher propensity for risk, highlighting the need for targeted interventions.
• The presence and gender of passengers can significantly impact a teenager's driving speed, either by encouraging safe behavior or promoting risk-taking.
• Educational programs and policies can help mitigate risky driving behavior by addressing these influential factors.

Studying teenage driving behavior through statistical analysis provides a clearer picture, enabling the development of effective strategies to reduce accidents and promote safer driving habits.