Answer to Question #251274 in Statistics and Probability for John

Question

The mean lifetime of 200 florescent light tubes gave a mean lifetime of 1560 hours with a standard deviation of 50hours, is it likely that a sample have come from a population with a mean lifetime of 1500 hours?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=1500$

$H_1:\mu\not=1500$

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 $\alpha = 0.05, df=n-1$

$=200-1=199$ ​​degrees of freedom, and the critical value for a two-tailed test is$t_c = 1.971957.$

The rejection region for this two-tailed test is $R = \{t: |t| > 1.971957\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{1560-1500}{50/\sqrt{200}}=16.970563$$

Since it is observed that $|t|= 16.970563>1.971957=t_c$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value for two-tailed, $\alpha=0.05, df=199,$ $t=16.97$ is $p\approx0,$ and since $p=0<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is different than 1500, at the $\alpha = 0.05$ significance level.