Question

Tire Life Prompt: A major tire manufacturer claims their heavy-duty truck tires have an average usage life of 71,000 miles. The shipping department of the company you work for has been using these tires for several years and feels they are not getting the mileage promised. The manager pulled 25 maintenance records and found an average tire life of 68,050 miles, with a standard deviation of 11,602 miles. He asks you to conduct a test of hypothesis to determine if the actual life of the tires is less than the manufacturer's claim. Response Parameters Use what you have learned about hypothesis testing to answer the following questions. What type of test should you perform? Which of the three equations for hypothesis testing should you use? Why did you choose that one? You may assume tire life is normally distributed. State your null and alternate hypotheses. Why did you choose those values and mathematical operators? What is the value of your test statistic? (Clearly, show how you arrived at this value.) Interpret the test statistic: Choose an appropriate confidence level, then evaluate the test statistic using either the critical value or the p-value approach. Why did you choose the confidence level that you did? Clearly, state the outcome of your test of hypothesis. What does your outcome mean in statistical terms? What does your outcome mean in terms of the problem?

Answer 1

We will conduct t-test for mean, since the population standard deviation of the tire life is not mentioned in the question.

The provided sample mean is X = 68050 and the sample standard deviation is s = 11602, and the sample size is n = 25. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: u = 71000 Ha: u < 71000 This corresponds to a left-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used. (2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the critical value for a left- tailed test is te = -1.711. The rejection region for this left-tailed test is R=t:t< -1.711 (3) Test Statistics The t-statistic is computed as follows: t= X - po s/n 68050 - 71000 º = -1.271 11602/25 (4) Decision about the null hypothesis Since it is observed that t = -1.271 > te = -1.711, it is then concluded that the null hypothesis is not rejected. Using the P-value approach: The p-value is p = 0.1079, and since p=0.1079 > 0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean y is less than 71000, at the 0.05 significance level.

Since we fail to reject the null hypothesis, we have no evidence to say that the actual life of the tire is less than the manufacturer's claim