To calculate the confidence interval for the average maximum horsepower (HP) of the experimental engine, we can follow the process for a t-distribution since the sample size is small (n = 16) and we are using the sample standard deviation.

Given Data:

• Sample size $n = 16$
• Sample mean $\bar{x} = 630 \text{ HP}$
• Sample standard deviation $s = 50 \text{ HP}$
• Confidence level $1 - \alpha = 0.99$ (since $\alpha = 0.01$)

Step 1: Determine the t-critical value

Since we are using a significance level of $\alpha = 0.01$, the confidence level is 99%, and the degrees of freedom $df = n - 1 = 16 - 1 = 15$.

Using a t-table (or a calculator), the t-critical value for a 99% confidence interval with 15 degrees of freedom is approximately:

$t_{0.005, 15} \approx 2.947$

Step 2: Calculate the margin of error (E)

The margin of error $E$ is calculated using the formula:

$E = t_{\alpha/2} \times \left(\frac{s}{\sqrt{n}}\right)$

Substitute the given values:

$E = 2.947 \times \left(\frac{50}{\sqrt{16}}\right) = 2.947 \times \left(\frac{50}{4}\right) = 2.947 \times 12.5 = 36.8375$

Round to two decimal places:

$E \approx 36.84 \text{ HP}$

Step 3: Calculate the confidence interval

The confidence interval for the population mean is:

$\left( \bar{x} - E , \bar{x} + E \right) = \left( 630 - 36.84 , 630 + 36.84 \right)$

$= \left( 593.16 , 666.84 \right)$

Conclusion:

The 99% confidence interval for the average maximum HP of the experimental engine is (593.16 HP, 666.84 HP).

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Would you like further clarification on any steps?
Here are 5 related questions you could explore:

1. How does increasing the sample size affect the confidence interval?
2. What is the difference between using a t-distribution and a z-distribution?
3. Why do we use a t-distribution in this problem instead of a normal distribution?
4. How does changing the confidence level affect the width of the interval?
5. What would happen to the confidence interval if the standard deviation was larger?

Tip: As the sample size increases, the t-distribution approaches the normal distribution, making the t-critical value smaller.