Questions: A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8,000. A survey of owners of that tire design is conducted. Of the 32 tires surveyed, the mean lifespan was 46,700 miles with a standard deviation of 9,800 miles. Using alpha = 0.05, is the data highly consistent with the claim? Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) State the distribution to use for the test. (Round your answers to two decimal places.)

Transcript text: A particular brand of tires claims that its deluxe tire averages at least 50,000 miles before it needs to be replaced. From past studies of this tire, the standard deviation is known to be 8,000 . A survey of owners of that tire design is conducted. Of the 32 tires surveyed, the mean lifespan was 46,700 miles with a standard deviation of 9,800 miles. Using alpha $=0,05$, is the data highly consistent with the claim? Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) State the distribution to use for the test. (Round your answers to two decimal places.) \[ \bar{x} \sim(\square) \]

Solution

Solution Steps

Step 1: Standard Error Calculation

To calculate the standard error $ SE $, we use the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{9800}{\sqrt{32}} \approx 1414.2136 \]

Step 2: Test Statistic Calculation

The test statistic $ Z $ is calculated using the formula:

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{46700 - 50000}{1414.2136} \approx -2.3335 \]

Step 3: P-value Calculation

For a left-tailed test, the P-value is given by:

\[ P = T(z) \approx 0.0098 \]

Step 4: Conclusion

We compare the P-value with the significance level $ \alpha = 0.05 $:

Since $ P \approx 0.0098 < 0.05 $, we reject the null hypothesis. This indicates that the data is not consistent with the claim that the average lifespan of the tires is at least 50,000 miles.