Answer to Question #238134 in Statistics and Probability for Tony

Question

The mean lifetime of a sample of 144 fluorescent light bulbs produced by a company is computed to be 170 hours with standard deviation of 120 hours.Test the company's claim that its bulbs' mean time is 200 hours at 0.05 significance level

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=200$

$H_1:\mu\not=200$

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 $\alpha=0.05,$

$df=n-1=144-1=143$ degrees of freedom, and the critical value for a two-tailed test is $t_c=1.976692.$

The rejection region for this two-tailed test is $R=\{t:|t|>1.976692\}.$

The t-statistic is computed as follows:

$$t=\dfrac{x-\mu}{s/\sqrt{n}}=\dfrac{170-200}{120/\sqrt{144}}=-3$$

Since it is observed that $|t| = 3 > t_c = 1.976692=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value for two-tailed $\alpha=0.05, df=143,t=-3$ is $p=0.003186,$ and since $p=0.003186<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is different than 200, at the $\alpha=0.05$ significance level.