To construct a 90% confidence interval for the mean difference between the mileage from cars using regular and premium gas, we can proceed with the following steps:

1. Calculate the Differences: First, calculate the difference in mileage (Premium - Regular) for each car.

2. Find the Mean and Standard Deviation of the Differences: Compute the mean and standard deviation of these differences.

3. Calculate the Standard Error (SE) of the Mean Difference: Use the standard deviation of the differences divided by the square root of the sample size.

4. Determine the Critical t-value for a 90% Confidence Interval: Since the sample size is 10, we'll use $t_{0.05, 9}$, corresponding to 9 degrees of freedom.

5. Construct the Confidence Interval: Apply the formula:

   $\text{Mean Difference} \pm t \times \text{SE}$

Step-by-Step Calculation:

1. Differences (Premium - Regular):

   • $19 - 13 = 6$
   • $22 - 20 = 2$
   • $24 - 21 = 3$
   • $25 - 23 = 2$
   • $25 - 22 = 3$
   • $26 - 27 = -1$
   • $26 - 25 = 1$
   • $28 - 27 = 1$
   • $32 - 28 = 4$
   • $24 - 22 = 2$

   Differences: $6, 2, 3, 2, 3, -1, 1, 1, 4, 2$

2. Mean of the Differences:

   $\text{Mean} = \frac{6 + 2 + 3 + 2 + 3 - 1 + 1 + 1 + 4 + 2}{10} = \frac{23}{10} = 2.3$

3. Standard Deviation of the Differences:

   Using the formula for standard deviation:

   $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$

   After calculating, we find $s \approx 1.89$.

4. Standard Error of the Mean Difference (SE):

   $SE = \frac{s}{\sqrt{n}} = \frac{1.89}{\sqrt{10}} \approx 0.60$

5. Critical t-value (for 90% confidence with 9 degrees of freedom): Using a t-table, $t_{0.05, 9} \approx 1.833$.

6. Confidence Interval Calculation:

   $2.3 \pm 1.833 \times 0.60$

   This simplifies to:

   $2.3 \pm 1.10$

   So, the 90% confidence interval for the mean difference is approximately:

   $(1.20, 3.40)$

Answer:

The 90% confidence interval estimate for the mean difference between the mileage from cars using regular and premium gas is (1.20, 3.40).

Would you like further details or have any questions?

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Here are five related questions for deeper understanding:

1. How would the confidence interval change with a larger sample size?
2. What would a 95% confidence interval for the mean difference look like?
3. How does the assumption of normality in differences affect this interval?
4. What if one type of gas consistently yielded higher mileage—how would that influence interpretation?
5. How would results differ if we had paired data over multiple days instead of a single day?

Tip: Confidence intervals widen with smaller sample sizes or higher confidence levels due to increased uncertainty.