91Ó°ÊÓ

Some people believe that talking on a cell phone while driving slows reaction time, increasing the risk of accidents. The study described in the paper "A Comparison of the Cell Phone Driver and the Drunk Driver" (Human Factors [2006]: \(381-391\) ) investigated the braking reaction time of people driving in a driving simulator. Drivers followed a pace car in the simulator, and when the pace car's brake lights came on, the drivers were supposed to step on the brake. The time between the pace car brake lights coming on and the driver stepping on the brake was measured. Two samples of 40 drivers participated in the study. The 40 people in one sample used a cell phone while driving. The 40 people in the second sample drank a mixture of orange juice and alcohol in an amount calculated to achieve a blood alcohol level of \(0.08 \%\) (a value considered legally drunk in most states). For the cell phone sample, the mean braking reaction time was 779 milliseconds and the standard deviation was 209 milliseconds. For the alcohol sample, the mean breaking reaction time was 849 milliseconds and the standard deviation was \(228 .\) Is there convincing evidence that the mean braking reaction time is different for the population of drivers talking on a cell phone and the population of drivers who have a blood alcohol level of \(0.08 \%\) ? For purposes of this exercise, you can assume that the two samples are representative of the two populations of interest.

Step by step solution

01

Set up the hypotheses

The null hypothesis (\(H_0\)) states that the mean braking reaction time is the same for all drivers, regardless of whether they are talking on a cell phone or have a blood alcohol level of 0.08%. Mathematically, this can be represented as: \(H_0: \mu_1 = \mu_2\)The alternative hypothesis (\(H_1\)) states that the mean braking time varies between the two groups. This can be represented as: \(H_1: \mu_1 \neq \mu_2\) where \(\mu_1\) and \(\mu_2\) are the mean braking times for the cell phone group and alcohol group respectively.

02

Calculate the test statistic

The test statistic for this two-sample t-test can be calculated using the formula: \[t = \frac{\(\bar X_1 - \bar X_2\)}{sqrt{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}\]where \(\bar X_1\) and \(\bar X_2\) are the sample means, \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes. Substituting the given values and performing the necessary calculations, we get the value of the test statistic.

03

Determine Degrees of Freedom and Find the Critical Value

Degrees of freedom for this two-sample t-test are calculated using the formula: \(df = n_1 + n_2 - 2 = 40 + 40 - 2 = 78\). As we are conducting a two-tailed test (as the alternative hypothesis is interested in difference, not a specific direction), we need to find the critical t-value that corresponds to an alpha level of 0.05 split between two tails (0.025 in each tail) on a t-distribution with 78 degrees of freedom.

04

Make a decision

Assuming we use a significance level of 5% (or \(\alpha = 0.05\)), If our calculated test statistic is greater than the critical t-value or less than the negative of the critical value, we reject the null hypothesis. If it is less than the critical t-value and greater than the negative of the critical value, we fail to reject the null hypothesis.

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.

Two-Sample t-Test

A two-sample t-test is used to compare the means from two different groups to see if they are significantly different from each other. In our exercise, the two groups are drivers using a cell phone and drivers with a blood alcohol level of 0.08%. The goal is to check if there is a significant difference in braking reaction times between these groups.

The two-sample t-test uses the means and standard deviations of the two groups, along with sample sizes, to calculate a test statistic. This statistic helps us determine if the observed differences in means are likely due to random chance or if there is indeed an actual difference. In simple terms, it checks whether the averages of two sets are significantly different, considering the variations in data and sample sizes.

Using the calculated test statistic, we can compare it against a critical value from the t-distribution to make our inference. If the test statistic is significantly beyond the critical values, it suggests that the mean braking times between drivers are different, potentially impacting driving safety.

Null and Alternative Hypotheses

In hypothesis testing, we always start with forming the null and alternative hypotheses. The null hypothesis (\(H_0\)) is a statement of no effect or no difference, which we aim to test against with the alternative hypothesis (\(H_1\)).

For our exercise, the null hypothesis proposes that there is no difference in the mean braking reaction times between drivers using a cell phone and those with a blood alcohol level of 0.08%, represented as \(H_0: \mu_1 = \mu_2\). This assumes that any difference observed is merely due to sampling variability.

Conversely, the alternative hypothesis suggests that there is a significant difference between the two means, represented as \(H_1: \mu_1 eq \mu_2\). The alternative hypothesis targets the presence of effect, implying that the conditions of driving (cell phone use vs. alcohol intake) do influence reaction times significantly.

Significance Level and Critical Value

The significance level, often denoted as \(\alpha\), plays a crucial role in hypothesis testing, setting the threshold for deciding whether to reject the null hypothesis. In our example, a significance level of 0.05 (or 5%) is commonly used, meaning we accept a 5% chance of falsely rejecting the null hypothesis.

The critical value, on the other hand, is a cut-off point determined by the significance level that defines the region where we would reject the null hypothesis. It depends on the degrees of freedom, which in this case is calculated as the total number of data points from both groups minus two. For our study, with 78 degrees of freedom, the critical value is from a t-distribution with 0.025 in each tail, due to the two-tailed nature of our test.

If the test statistic calculated from the data exceeds the critical value or is less than the negative critical value, we reject \(H_0\) and accept \(H_1\), suggesting significant differences in reaction times between the groups. This threshold ensures that our conclusion is made within an acceptable level of uncertainty.