High-power experimental engines are being developed by the Stevens Motor Company for use in its new sports coupe. The engineers have calculated the maximum horsepower for the engine to be 600HP
. Sixteen engines are randomly selected for horsepower testing. The sample has an average maximum HP of 630
 with a standard deviation of 50HP
. Assume the population is normally distributed.
Step 1 of 2 :  Calculate a confidence interval for the average maximum HP for the experimental engine. Use a significance level of α=0.01
. Round your answers to two decimal places. 

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)**.

---

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.

High-power experimental engines are being developed by the Stevens Motor Company for use in its new sports coupe. The engineers have calculated the maximum horsepower for the engine to be 600HP. Sixteen engines are randomly selected for horsepower testing. The sample has an average maximum HP of 630 with a standard deviation of 50HP. Assume the population is normally distributed. Step 1 of 2: Calculate a confidence interval for the average maximum HP for the experimental engine. Use a significance level of α=0.01. Round your answers to two decimal places.

Calculate the 99% confidence interval for the average maximum horsepower of Stevens Motor Company's experimental engine. This step-by-step solution covers statistical concepts such as confidence intervals and t-distribution, using sample data with a mean of 630 HP and standard deviation of 50 HP. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation