Answer to Question #345869 in Statistics and Probability for gela

Question

A manufacturer of ball pens claims that a certain pen they manufactures has a mean writing life of 400 pages. A purchasing agent selects a sample of 100 pens and puts them for test. The mean writing life from the sample was 390 pages with a standard deviation of 30 at alpha 0.01.

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 $\alpha = 0.01,$ $df=n-1=99$ and the critical value for a two-tailed test is $t_c = 2.626405.$

The rejection region for this two-tailed test is $R = \{t:|t|> 2.626405\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{390-400}{30/\sqrt{100}}=-3.3333$$

Since it is observed that $|t|=3.3333> 2.626405=t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for two-tailed, $df=99$ degrees of freedom, $t=-3.3333$ is $p=0.001208,$ and since $p=0.001208<0.01=\alpha,$ 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 $\alpha = 0.01$ significance level.