Question

a light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 765 hours. a random sample of 23 light bulbs has a mean life of 749 hours. assume the population is normally distributed and the population standard deviation is 58 hours. at \\( \alpha = 0.05 \\), do you have enough evidence to reject the manufacturers claim? complete parts (a) through (e).
(c) identify the standardized test statistic. use technology.
\\( z = - 1.32 \\) (round to two decimal places as needed.)
(d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.
a. fail to reject \\( h _ { 0 } \\). there is not sufficient evidence to reject the claim that mean bulb life is at least 765 hours.
b. reject \\( h _ { 0 } \\). there is not sufficient evidence to reject the claim that mean bulb life is at least 765 hours.
c. fail to reject \\( h _ { 0 } \\). there is sufficient evidence to reject the claim that mean bulb life is at least 765 hours.
d. reject \\( h _ { 0 } \\). there is sufficient evidence to reject the claim that mean bulb life is at least 765 hours.

Explanation:

Step1: Determine the critical value

Since the significance level \(\alpha = 0.05\) and this is a left - tailed test (because the claim is \(\mu\geq765\) and we are testing against the alternative \(\mu < 765\)), the critical value \(z_{\alpha}\) is \(z_{0.05}=- 1.645\) (from the standard normal distribution table).

Step2: Compare the test statistic and the critical value

The test statistic \(z=-1.32\). We know that if \(z>z_{\alpha}\) (in a left - tailed test), we fail to reject the null hypothesis. Here, \(-1.32>-1.645\) (i.e., the test statistic is not in the rejection region).

Answer:

A. Fail to reject \(H_{0}\). There is not sufficient evidence to reject the claim that mean bulb life is at least 765 hours.