Answer to Question #316516 in Statistics and Probability for secret

Question

An electrical firm produces light bulbs that have a length of life that is approximately normally distributed with a mean of 650 hours and a population standard deviation of 20 hours. A new version of light bulbs is being produced and is assumed to be better than the previous version. To test this claim, a random sample of 52 new light bulbs are tested. Would you agree with this claim if the random sample showed an average of 760 hours? Use a 0.01 level of significance.

Answer 1

$\mu=650, \sigma=20,n=52,\bar{x}=760,\alpha=0.01.$

Null and alternative hypotheses:

$H_0:\mu\leq650;\\ H_1:\mu>650.$

Because $\sigma$ is known and $n=52>30,$ we can use the z-test.

$z=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\cfrac{760-650}{20/\sqrt{52}}=39.66.$

In z-table, the area corresponding to $z=39.66$ is 1. Because the test is a right-tailed test, the P-value is equal to the area to the right of $z=39.66,$ so, $P=1-1=0.$

Because the P-value is less than $\alpha$ =0.01, we reject the null hypothesis, there is enough evidence at the 1% level of significance to support the claim that the new version of light bulbs to be better than the previous version (the mean of length of life is more than 650 hours).