Answer to Question #334218 in Statistics and Probability for Mashon

Question

The CEO of a battery manufacturing company claimed that their batteries would last an average of 280 hours under normal use. A researcher randomly selected 20 batteries from the production line and tested them. The tested batteries had a mean life span of 250 hours with a standard deviation of 40 hours. Do we have enough evidence to suggest that the claim of an average of 280 hours is false?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=280$

$H_1:\mu\not=280$

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 $\alpha = 0.05,$ $df=n-1=20-1=19$ degrees of freedom, and the critical value for a two-tailed test is $t_c = 2.093024$

The rejection region for this two-tailed test is$R = \{t: |t| > 2.023024\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{250-280}{40/\sqrt{20}}\approx-3.354102$$

Since it is observed that $|t| = 3.354102 >2.093024= t_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for two-tailed, $df=19$ degrees of freedom, $t=-3.354102$ is $p = 0.003333,$ and since $p=0.003333<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is different than 280, at the $\alpha = 0.05$ significance level.