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The problem involves a hypothesis test to determine whether there is a significant difference in the braking distances of two different models (Model A and Model B) of a high-end sedan. The braking distances for six drivers are provided, and the test is conducted at a significance level \(\alpha = 0.01\).

### Step 2 of 3: Compute the value of the test statistic

We can approach this problem as a paired-sample (dependent sample) t-test, as the same drivers tested both models. The key steps for calculating the t-test statistic are:

1. **Calculate the differences** between braking distances for each driver (Model A - Model B).
2. **Find the mean of these differences** (\(\bar{d}\)).
3. **Calculate the standard deviation of the differences** (\(s_d\)).
4. **Compute the test statistic** using the formula:
   \[
   t = \frac{\bar{d}}{s_d / \sqrt{n}}
   \]
   where \(n\) is the number of paired observations (in this case, 6).

Let me calculate these values for you.The calculations give the following results:

- Mean of the differences (\(\bar{d}\)): \(-2.5\)
- Standard deviation of the differences (\(s_d\)): \(2.258\)
- Test statistic (\(t\)): \(-2.712\)

This test statistic will be compared to the critical value from the t-distribution at \(\alpha = 0.01\) with \(n - 1 = 5\) degrees of freedom to draw a conclusion.

Would you like me to continue with the hypothesis testing to see if the results are statistically significant? Also, let me know if you have any questions about the computations.

Here are 5 related questions to expand on this topic:
1. How do we find the critical t-value for a paired t-test?
2. What is the interpretation of the p-value in hypothesis testing?
3. How does the significance level (\(\alpha\)) affect the conclusion of the test?
4. What assumptions must be satisfied to perform a paired t-test?
5. How would you interpret the result if the test was not significant?

**Tip:** In hypothesis testing, always verify if the assumptions of the test (e.g., normality of differences) are met to ensure accurate conclusions.

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 measured in feet. Can the auto dealer conclude that there is a significant difference in the braking distances of the two models? Use α = 0.01.

Determine if there is a significant difference in the braking distances between two models of high-end sedans using a paired sample t-test. The problem involves hypothesis testing at a significance level of α = 0.01, with braking data collected from six drivers.

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