Question

An automobile dealer believes that his new model will give trouble-free service for at least 15,000 miles. In a test of 8 new models, the following number of trouble-free miles were recorded:

16,000 15,000 14,350 14,500 13,750 13,000 14,250 13,900

Do the data support the automobile dealer's claim. Let α=0.01

Answer 1

The hypothesis test fails to reject the null hypothesis, so there is not enough evidence to conclude that the mean trouble-free miles is less than 15,000.

Step-by-step explanation:

To determine if the data supports the automobile dealer's claim, we can perform a hypothesis test. The null hypothesis (H0) is that the mean trouble-free miles is at least 15,000, while the alternative hypothesis (Ha) is that the mean trouble-free miles is less than 15,000. We can use a one-sample t-test with a significance level of α=0.01 to evaluate the evidence.

To perform the hypothesis test, we calculate the sample mean and standard deviation of the recorded trouble-free miles. The sample mean is 14,452.5 miles and the sample standard deviation is 661.70 miles. We then calculate the t-statistic, which is given by (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size)). Plugging in the values, we get t = (14,452.5 - 15,000) / (661.70 / sqrt(8)) = -1.052. We then compare the t-statistic to the critical t-value from the t-distribution table with 7 degrees of freedom.

If the absolute value of the t-statistic is greater than the critical t-value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis. Using a significance level of α=0.01, the critical t-value is -3.499. Since the absolute value of the t-statistic (-1.052) is less than the critical t-value, we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the mean trouble-free miles is less than 15,000.