Answer to Question #218079 in Statistics and Probability for Ash

Question

A manufacturer of fluorescent tubes claims that average life rime is more than 1750 hour, A sample of 400 fluorescent light bulbs produced mean life tone 1600 hours with a standard deviation of ISO hours. Does the claim of the company justified at 5% level of significance?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu\leq 1570$

$H_1:\mu>1570$

This corresponds to a right-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 $\alpha=0.05, df=n-1$

$=400-1=399$ degrees of freedom, and the critical value for a right-tailed test is $t_c=1.648682.$

The rejection region for this right-tailed test is $R=\{t:T>1.648682\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{1600-1570}{150/\sqrt{400}}=4$$

Since it is observed that $t=4>1.648682=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value for right-tailed $\alpha=0.05, df=399, t=4$ is

$p=0.000038,$ and since $p=0.000038<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$  is greater than $1570,$ at the $\alpha=0.05$ significance level.