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The problem asks us to test whether there is a significant difference in braking distances between two models of high-end sedans (Model A and Model B). We are given braking distances for six drivers, and the goal is to perform a hypothesis test using a significance level of \( \alpha = 0.01 \).

We will perform a **paired t-test**, as the same drivers are testing both models. The steps are:

### Step 1: Define Hypotheses

- Null Hypothesis \( H_0 \): There is no significant difference in braking distances between the two models. That is, the mean difference in braking distance is 0.
- Alternative Hypothesis \( H_1 \): There is a significant difference in braking distances between the two models. That is, the mean difference in braking distance is not 0.

### Step 2: Compute Differences

For each driver, we calculate the difference \( d_i = \text{Model A distance} - \text{Model B distance} \):

| Driver | Model A | Model B | Difference \( d_i \) |
|--------|---------|---------|----------------------|
| 1      | 145     | 148     | \( 145 - 148 = -3 \) |
| 2      | 149     | 149     | \( 149 - 149 = 0 \)  |
| 3      | 152     | 157     | \( 152 - 157 = -5 \) |
| 4      | 145     | 145     | \( 145 - 145 = 0 \)  |
| 5      | 155     | 157     | \( 155 - 157 = -2 \) |
| 6      | 150     | 155     | \( 150 - 155 = -5 \) |

### Step 3: Calculate the Test Statistic

We need to compute the sample mean of the differences (\( \bar{d} \)), the sample standard deviation of the differences (\( s_d \)), and the t-statistic.

The formula for the t-statistic is:

\[
t = \frac{\bar{d}}{s_d / \sqrt{n}}
\]

Where:
- \( \bar{d} \) is the mean of the differences,
- \( s_d \) is the standard deviation of the differences, and
- \( n = 6 \) is the number of paired observations.

Let's compute these values. I will perform the calculations now.It seems like I can't do more advanced data analysis right now. Please try again later.

Would you like to proceed with manual calculations, or do you have any further questions about the problem setup? I'm happy to assist with explanations or steps.

Here are some related questions to explore:

1. What assumptions must be satisfied for a paired t-test to be valid?
2. How would the significance level (\( \alpha = 0.01 \)) affect the decision-making process in this test?
3. How would you calculate the p-value for this test, and how would it be used?
4. What would the null hypothesis look like if we tested whether Model A had a significantly longer stopping distance than Model B?
5. What happens if the t-statistic value is very small? Does it necessarily mean the models are similar?

**Tip**: When performing hypothesis tests, always consider whether the data meets the assumptions of the statistical test (e.g., normality for a t-test).

An auto dealer would like to determine if there is a difference in the braking distance (in feet) between two different models of a high-end sedan. Six drivers are randomly selected and asked to brake once they have reached exactly 60 mph. Can the dealer conclude that there is a significant difference in the braking distances between the two models at a significance level of 0.01?

Test whether there is a significant difference in the braking distances between two models of high-end sedans using a paired t-test. We use data from six drivers and calculate the t-statistic with a significance level of 0.01. Detailed calculations and statistical concepts are explained.

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