Question

To claim a tire under warranty, a 19.5" size tire cannot be more than four years old and 16" size or smaller tires cannot be more than six years old.

Answer 1

The question involves a hypothesis test (Z-test) to determine if the mean lifespan of tires, as reported from a survey, is highly inconsistent with the claimed average lifespan. We calculate the test statistic using the sample mean, claimed mean, standard deviation, and sample size, and compare it to the critical Z value at alpha 0.05.

Step-by-step explanation:

To assess whether the data is highly inconsistent with the claim that the deluxe tire averages at least 50,000 miles before replacement, we use a hypothesis test. The null hypothesis (H0) is that the mean lifespan of the tires is at least 50,000 miles, against the alternative hypothesis (Ha) that the mean lifespan is less than 50,000 miles.

Since the standard deviation of the population is known, we can perform a Z-test. First, we calculate the test statistic using the formula:

Z = (mean - claimed mean) / (standard deviation / sqrt(sample size))

Plugging in the values, we get:

Z = (46,500 - 50,000) / (8,000 / sqrt(28))

Using standard statistical tables for the normal distribution, we compare the calculated Z value to the Z value for a significance level of alpha 0.05. If the calculated Z value is beyond the critical value, the null hypothesis is rejected, indicating that the data is highly inconsistent with the claim.

If we find the Z value is not past the critical threshold, then we would not reject the null hypothesis, meaning the data is not highly inconsistent with the claim.

Answer 2

The question is about using a t-test to analyze if the average tire lifespan from a sample is significantly different from the claimed lifespan. It involves statistics, applying hypothesis testing at an alpha level of 0.05, and assessing the consistency of sample data with a given claim.

Step-by-step explanation:

The subject of this question is Mathematics, specifically the area of statistics within the context of hypothesis testing. The student is tasked with determining whether the sample data of a tire's lifespan is highly inconsistent with the claimed average lifespan of the tire using a specified alpha level of 0.05.

To do this, one would typically employ a t-test for the mean, considering the sample size is less than 30 and the population standard deviation is not known.

To conduct this analysis, the null hypothesis (H0) would suggest that the true mean mileage of the tires is at least 50,000 miles. The alternative hypothesis (H1) would indicate that the true mean mileage is less than 50,000 miles.

With the sample data provided (mean = 46,500 miles, sample standard deviation = 9,800 miles, n=28), and using the alpha level of 0.05, we would calculate the t-statistic and compare it against the critical t-value for 27 degrees of freedom (n-1).

If the calculated t-statistic falls within the critical region (in the lower tail of the distribution since it's a one-tailed test), the data would be considered highly inconsistent with the claim, leading to the rejection of the null hypothesis.