Questions: Correct An auto dealer would like to determine if there is a difference in the braking distance (the number of feet required to go from 60 mph to 0 mph) of two different models of a high-end sedan. Six drivers are randomly selected and asked to drive both models and brake once they have reached exactly 60 mph. The distance required to come to a complete halt is then measured in feet. The results of the test are as follows. Can the auto dealer conclude that there is a significant difference in the braking distances of the two models? Use α=0.01. Let the braking distances of Model A represent Population 1 and the braking distances of Model B represent Population 2. Braking Distance of High-End Sedans (Feet): Driver 1 2 3 4 5 6 Model A 151 145 150 158 158 151 Model B 153 145 151 160 159 151

Solution Steps

To determine if there is a significant difference in the braking distances of the two models, we can perform a paired t-test. This test is appropriate because the same drivers tested both models, making the data paired. The steps are as follows:

1. Calculate the differences between the braking distances for each driver.
2. Compute the mean and standard deviation of these differences.
3. Use the mean and standard deviation to calculate the t-statistic.
4. Compare the t-statistic to the critical value from the t-distribution with \( n-1 \) degrees of freedom, where \( n \) is the number of pairs, to determine if the difference is statistically significant at the given significance level (\(\alpha = 0.01\)).

For each driver, calculate the difference in braking distances between Model A and Model B. The differences are: \[ \text{differences} = [151 - 153, 145 - 145, 150 - 151, 158 - 160, 158 - 159, 151 - 151] = [-2, 0, -1, -2, -1, 0] \]

Calculate the mean (\(\bar{d}\)) and standard deviation (\(s_d\)) of the differences: \[ \bar{d} = \frac{-2 + 0 - 1 - 2 - 1 + 0}{6} = -1.0 \] \[ s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}} = 0.8944 \]

Using the mean and standard deviation of the differences, calculate the t-statistic: \[ t = \frac{\bar{d}}{s_d / \sqrt{n}} = \frac{-1.0}{0.8944 / \sqrt{6}} = -2.7386 \]

Final Answer