Question

A trucking firm suspects the claim that the average lifetime of

certain tires is 28, 000 miles. To check the claim, the firm puts 40

of these tires on its trucks and gets a mean lifetime of 27, 463

miles with a standard deviation of 1, 348 miles. Can we accept the

claim at the significance level = 0.01 (i.e. the type I error is to

be at most 0.01) using rejection region

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0: \mu=28000$

$H_1: \mu\not=28000$

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha=0.01,$

$df=n-1=40-1=39$ degrees of freedom, and the critical value for a two-tailed test is $t_c=2.707913.$

The rejection region for this two-tailed test is $R=\{t: |t|>2.707913\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{27463-28000}{1348/\sqrt{40}}\approx-2.519500$$

Since it is observed that $|t|=2.519500<2.707913,$ it is then concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$ is different than $28000,$ at the $\alpha=0.01$ significance level.

Using the P-value approach: The p-value for two-tailed $\alpha=0.01, df=39,$

$t=-2.519500$ is $p=0.01596,$ and since $p=0.01596>0.01=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$ is different than $28000,$ at the $\alpha=0.01$ significance level.

Therefore, there is not enough evidence to claim that the average lifetime of certain tires is is different than $28000$ miles, at the $\alpha=0.01$ significance level.