Answer to Question #344197 in Statistics and Probability for ade

Question

The mean lifetime of the light bulbs produced by Volta Corporation is 1705 hours with a population standard  deviation of 150 hours. The CEO of the company claims that a new production process has led to an increase in  the mean lifetimes of the light bulbs. If Jeremy tested 130 bulbs made from the new production process and found  that their mean lifetime is 1650 hours, test the hypothesis that the mean is not equal to 1705 hours using the level of significance of 0.10.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=1705$

$H_1:\mu\not=1705$

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Based on the information provided, the significance level is $\alpha = 0.10,$ and the critical value for a two-tailed test is $z_c = 1.6449.$

The rejection region for this two-tailed test is $R = \{z:|z|>1.6449\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{1650-1705}{150/\sqrt{130}}\approx-4.1806$$

Since it is observed that $|z|=4.1806>1.6449=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=2P(z<-4.1806)= 0.000029,$ and since $p= 0.000029<0.10=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$

is different than 1705, at the $\alpha = 0.10$ significance level.