Answer to Question #163131 in Statistics and Probability for Juju

Question

A factory manufactures cars with a warranty of 5 years on the engine and transmission.  An engineer believes that the engine or transmission will malfunction in less than 5 years.  He tests a sample of 40 cars and find the average time to be 4.8 years with a standard deviation of 0.50. At a 2% confidence level, is there enough evidence to support the idea that the warranty should be revised ?  ( P-value Test)

Answer 1

The provided sample mean is $\bar{x}=4.8$ and the sample standard deviation is $s=0.50.$

The sample size is $n=40.$ The number of degrees of freedom are $df=n-1=39,$ and the significance level is $\alpha=0.02.$

The following null and alternative hypotheses need to be tested:

$H_0: \mu\geq5$

$H_1: \mu<5$

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

Based on the information provided, the critical value for a left-tailed test is $t_c=-2.124742.$

The rejection region for this two-tailed test is $R=\{t:t<-2.124742\}$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{4.8-5}{0.5/\sqrt{40}}=-2.529822$$

Since it is observed that $t=-2.529822<-2.124742=t_c,$ it is then concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 5, at the 0.02 significance level.

Using the P-value approach: The p-value $df=39, t=-2.529822,$ $left-tailed,$ is $p=0.00778210,$ and since $p=0.00778210<0.02,$ it is concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 5, at the 0.02 significance level.

Therefore there is enough evidence to support the idea that the warranty should be revised at the 0.02 significance level.