Question

A manufacturer of fluorescent bulbs claims that mean life of fluorescent bulb is 400 hours. a sample of 35 fluorescent bulbs was taken and it showed a mean of 399 hours with a standard deviation of 1.1 hours, is the mean life different from 400 hours? use the 0.05 significance level.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=400$

$H_1:\mu\not=400$

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 $\alpha=0.05, df=n-1$

$=35-1=34$ ​degrees of freedom, and the critical value for a two-tailed test is$t_c= 2.032244.$

The rejection region for this two-tailed test is $R=\{t:|t|> 2.032244\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{399-400}{1.1/\sqrt{35}}=-5.378254$$

Since it is observed that $|t|=5.378254>2.032244=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value for two-tailed, $\alpha=0.05, df=34,$ $t=-5.378254$ is $p=0.000006,$ and since $p=0.000006<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is different than 400, at the $\alpha=0.05$ significance level.