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To construct a 99% confidence interval for the mean difference in fuel economy, we need to calculate the differences for each pair of observations between Tread A and Tread B, then find the mean and standard deviation of these differences. Here’s the step-by-step approach:

1. **Calculate the Differences** \( d \) for each car:
   \[
   d = \text{Mileage of Tread A} - \text{Mileage of Tread B}
   \]

2. **Compute the Mean and Standard Deviation of \( d \):**
   - Calculate the mean (\( \bar{d} \)) of these differences.
   - Calculate the standard deviation (\( s_d \)) of these differences.

3. **Determine the Critical t-value** for a 99% confidence interval with \( n-1 \) degrees of freedom (in this case, \( n = 9 \)).

4. **Calculate the Margin of Error (ME)** using:
   \[
   \text{ME} = t_{\alpha/2} \cdot \frac{s_d}{\sqrt{n}}
   \]

5. **Construct the Confidence Interval**:
   \[
   \bar{d} - \text{ME} < \mu_d < \bar{d} + \text{ME}
   \]

I’ll perform the calculations for each step.The 99% confidence interval for the mean difference in fuel economy between Tread A and Tread B is:

\[
0.32 < \mu_d < 3.85
\]

This interval suggests we are 99% confident that the mean difference in mileage (in miles per gallon) between Tread A and Tread B lies between 0.32 and 3.85 miles per gallon, with Tread A having the higher mileage on average.

Would you like more details on the calculations, or do you have any questions?

Here are five related questions for further exploration:
1. What would the confidence interval look like at a 95% confidence level?
2. How does increasing the sample size affect the width of the confidence interval?
3. Why is the t-distribution used instead of the normal distribution for small sample sizes?
4. What assumptions are made when calculating a confidence interval for the mean difference?
5. How would you interpret this interval in a practical decision-making scenario?

**Tip:** Confidence intervals provide a range that likely includes the true mean difference, but they do not guarantee that every repeated interval would include the mean.

Construct a 99% confidence interval for the mean difference in fuel economy between Tread A and Tread B, based on mileage data for 9 cars.

Learn how to calculate a 99% confidence interval for the mean difference in fuel economy between two tire tread patterns. Using mileage data from 9 cars, this solution explores statistical concepts including mean difference, standard deviation, and margin of error, providing a step-by-step guide suitable for undergraduate-level statistics.

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