Answer to Question #206476 in Statistics and Probability for john

Question

The mean lifetime for a sample of 125 lamps is 1205 hours with standard deviation 105 hours. However, the company claims that their lamps average lifetime is difference from 1300 hours. Test the claim at 1% level of significance.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=1300$

$H_1:\mu\not=1300$

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 $\alpha=0.01,$ and the critical value for a two-tailed test for $\alpha=0.01$ and $df=n-1=125-1=124$ degrees of freedom is $t_c=2.61606.$ The rejection region for this two-tailed test is $R=\{t: |t|>2.61606\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{1205-1300}{105/\sqrt{125}}\approx-10.11555$$

Since it is observed that $|t|=10.11555>2.61606=t_c,$ it is then concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the population mean $\mu$ is different than 1300, at the $\alpha=0.01$ significance level.

Using the P-value approach: The p-value for two-tailed, $t=-10.11555, df=124,$ $\alpha=0.01$ is $p=0,$ and since $p=0<0.01,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is different than 1300, at the $\alpha=0.01$ significance level.