Question

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 29 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.) Part f. What is the p-value? (Round your answer to four decimal places.)

Answer 1

To test if the data is highly consistent with the claim, we can perform a hypothesis test. The null hypothesis is that the mean lifespan of the tires is greater than or equal to 50,000 miles. The alternative hypothesis is that the mean lifespan of the tires is less than 50,000 miles. Using alpha = 0.05, we find that the p-value is approximately 0.0362.

Step-by-step explanation:

In this problem, we want to determine if the data is highly consistent with the claim made by a tire brand. We are given that the claim is that the deluxe tire averages at least 50,000 miles before it needs to be replaced. We are also given that the sample mean lifespan of the tires is 46,700 miles with a standard deviation of 9,800 miles.

To test if the data is highly consistent with the claim, we can perform a hypothesis test. The null hypothesis, denoted as H0, is that the mean lifespan of the tires is greater than or equal to 50,000 miles. The alternative hypothesis, denoted as Ha, is that the mean lifespan of the tires is less than 50,000 miles.

Using alpha = 0.05, we can calculate the t-statistic and the p-value. In this case, the t-statistic is calculated as (sample mean - claimed mean) / (sample standard deviation/sqrt (n)) = (46,700 - 50,000) / (9,800 / sqrt(29)) = -1.9193. Using the t-distribution with 28 degrees of freedom, we can find the p-value associated with a t-statistic of -1.9193. The p-value turns out to be approximately 0.0362.

Since the p-value (0.0362) is less than the alpha value (0.05), we reject the null hypothesis. This means that there is sufficient evidence to conclude that the mean lifespan of the tires is less than 50,000 miles. Therefore, the data is highly inconsistent with the claim made by the tire brand.