Answer to Question #91115 in Statistics and Probability for Maxine

Question

a light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 720 hours.a random sample of 51 light bulbs as a mean of 712.8 hours with a population standard deviation of 62 hours at and a equals 0.05 can you support the company's claim using the test statistics?

Answer 1

Identify the null hypothesis and alternative hypothesis:

$$H_0: \mu\geq 720 \ (claim) (Bulbs \ last\ at\ least \ 720\ hours)$$ $$H_1: \mu<720 \ (Bulbs\ last\ less\ than\ 720\ hours)\ \ \ \ \ \ \ \ \ \$$

Identify the critical value(s):

Left-tailed test

$$z_\alpha=z_{0.05}=-1.645\approx-1.65$$

 The critical region will be to the left of –1.65.

Identify the standardized test statistic

$$z={\bar{x}-\mu \over \sigma/\sqrt{n} }={712.8-720 \over 62/\sqrt{51} }\approx-0.83$$

Decide whether to reject or fail to reject the null hypothesis and interpret the decision in the context of the original claim

A one-tailed test (right or left) indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean.

The rejection zone is below -1.65 at the 0.05 level.

Fail to reject $H_0.$ There is not sufficient evidence to reject the claim that mean bulb life is at least 720 hours.

A type I error occurs if one rejects the null hypothesis when it is true.

Type I Error: Incorrectly claim light bulbs last less than 720 hours

A type II error occurs if one does not reject the null hypothesis when it is false.

Type II Error: Fail to detect that light bulbs last less than 720 hours