Questions: A consumer organization wants to estimate the actual tread wear index of a brand name of tires that claims "graded 200" on the sidewall of the tire. A random sample of n=22 indicates a sample mean tread wear index of 198.3 and a sample standard deviation of 18.6. Complete parts (a) through (b). a. Assuming that the population of tread wear indexes is normally distributed, construct a 90% confidence interval estimate of the population mean tread wear index for tires produced by this manufacturer under this brand name. ≤ μ ≤ (Round to two decimal places as needed.) b. Do you think that the consumer organization should accuse the manufacturer of producing tires that do not meet the performance information on the sidewall of the tire? Explain. A. No, because a grade of 200 is in the interval. B. Yes, because a grade of 200 ists the interval. C. No, because a grade of 200 is not in the interval. D. Yes, because a grade of 200 is not in the interval.

A consumer organization wants to estimate the actual tread wear index of a brand name of tires that claims "graded 200" on the sidewall of the tire. A random sample of n=22 indicates a sample mean tread wear index of 198.3 and a sample standard deviation of 18.6. Complete parts (a) through (b). a. Assuming that the population of tread wear indexes is normally distributed, construct a 90% confidence interval estimate of the population mean tread wear index for tires produced by this manufacturer under this brand name. ≤ μ ≤ (Round to two decimal places as needed.) b. Do you think that the consumer organization should accuse the manufacturer of producing tires that do not meet the performance information on the sidewall of the tire? Explain. A. No, because a grade of 200 is in the interval. B. Yes, because a grade of 200 ists the interval. C. No, because a grade of 200 is not in the interval. D. Yes, because a grade of 200 is not in the interval.
Transcript text: A consumer organization wants to estimate the actual tread wear index of a brand name of tires that claims "graded 200 " on the sidewall of the tire. A random sample of $n=22$ indicates a sample mean tread wear index of 198.3 and a sample standard deviation of 18.6. Complete parts (a) through (b). a. Assuming that the population of tread wear indexes is normally distributed, construct a $90 \%$ confidence interval estimate of the population mean tread wear index for tires produced by this manufacturer under this brand name. $\leq \mu \leq$ $\square$ (Round to two decimal places as needed.) b. Do you think that the consumer organization should accuse the manufacturer of producing tires that do not meet the perfomance information on the sidewall of the tire? Explain. A. No, because a grade of 200 is in the interval. B. Yes, because a grade of 200 ists the interval. C. No, because a grade of 200 is not in the interval. D. Yes, because a grade of 200 is not in the interval.
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Solution

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Solution Steps

Step 1: Calculate the Confidence Interval

To estimate the population mean tread wear index, we construct a \(90\%\) confidence interval. Given that the sample size is \(n = 22\), the sample mean is \(\bar{x} = 198.3\), and the sample standard deviation is \(s = 18.6\), we use the formula for the confidence interval when the population variance is unknown:

\[ \bar{x} \pm t \frac{s}{\sqrt{n}} \]

For a \(90\%\) confidence level and \(n - 1 = 21\) degrees of freedom, the critical value \(t\) is approximately \(1.72\). Thus, the confidence interval is calculated as follows:

\[ 198.3 \pm 1.72 \cdot \frac{18.6}{\sqrt{22}} = 198.3 \pm 6.82 \]

This results in the confidence interval:

\[ (198.3 - 6.82, 198.3 + 6.82) = (191.48, 205.12) \]

Step 2: Evaluate the Manufacturer's Claim

The manufacturer claims a tread wear index of \(200\). We need to determine if this claimed value falls within the calculated confidence interval \((191.48, 205.12)\).

Since \(200\) is within the interval, the consumer organization should not accuse the manufacturer of producing tires that do not meet the performance information on the sidewall.

Final Answer

The \(90\%\) confidence interval for the population mean tread wear index is \(\boxed{(191.48, 205.12)}\).

The decision regarding the manufacturer's claim is \(\boxed{\text{A. No, because a grade of 200 is in the interval.}}\)

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