Question

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.

Short answer

Whether there's 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% depends on the comparison of calculated t-statistic with the critical t-value from the t-distribution table. If the computed t-statistic is in the critical region, we reject the null hypothesis indicating there's enough evidence to suggest a difference in mean reaction times.

Step-by-step solution

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.

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.

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.

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.