We are asked to construct a 90% confidence interval estimate for the population mean tread wear index, based on a sample of 22 tires. The formula for constructing a confidence interval for the population mean when the population standard deviation is unknown (and the sample size is less than 30) is:

$CI = \bar{x} \pm t_{\frac{\alpha}{2}, n-1} \left( \frac{s}{\sqrt{n}} \right)$

Where:

• $\bar{x}$ = sample mean = 240.1
• $s$ = sample standard deviation = 15.5
• $n$ = sample size = 22
• $t_{\frac{\alpha}{2}, n-1}$ = critical t-value for a 90% confidence interval and 21 degrees of freedom ($n-1$)

Step 1: Find the critical t-value

For a 90% confidence interval with 21 degrees of freedom, we need to find the t-value for $\alpha/2 = 0.05$. Using a t-distribution table or calculator, the critical value $t_{0.05, 21}$ is approximately 1.721.

Step 2: Calculate the margin of error (ME)

The margin of error is given by:

$ME = t_{\frac{\alpha}{2}, n-1} \left( \frac{s}{\sqrt{n}} \right)$

$ME = 1.721 \times \left( \frac{15.5}{\sqrt{22}} \right)$

Let's calculate the margin of error.The margin of error (ME) is approximately 5.69.

Step 3: Calculate the confidence interval

Now, we can calculate the confidence interval as:

$CI = \bar{x} \pm ME$

$CI = 240.1 \pm 5.69$

Thus, the confidence interval is:

$(240.1 - 5.69, 240.1 + 5.69) = (234.41, 245.79)$

Conclusion:

The 90% confidence interval for the population mean tread wear index is:

$234.41 \leq \mu \leq 245.79$

Would you like further details or have any other questions?

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Here are 5 related questions:

1. How does the confidence level affect the width of the confidence interval?
2. What is the significance of the sample size in confidence interval estimation?
3. How would the interval change if the sample standard deviation was larger?
4. What happens to the confidence interval if we increase the sample size?
5. How do we interpret a confidence interval in the context of this tire tread wear index study?

Tip: A larger sample size reduces the margin of error, leading to a narrower confidence interval, which provides a more precise estimate.