Question

a light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 774 hours. a random sample of 27 light bulbs has a mean life of 750 hours. assume the population is normally distributed and the population standard deviation is 65 hours. at \\( \alpha = 0.08 \\), do you have enough evidence to reject the manufacturers claim? complete parts (a) through (e). (claim) (claim) (claim) d. \\( h _ { 0 } : \mu \geq 774 \\) (claim) e. \\( h _ { 0 } : \mu < 750 \\) (claim) f. \\( h _ { 0 } : \mu \
eq 774 \\) (claim) \\( h _ { a } : \mu < 774 \\) \\( h _ { a } : \mu \geq 750 \\) \\( h _ { a } : \mu = 774 \\) (b) identify the critical value(s). use technology. \\( z _ { 0 } = - 1.41 \\) (use a comma to separate answers as needed. round to two decimal places as needed.) identify the rejection region(s). choose the correct answer below. a. b. c. (c) identify the standardized test statistic. use technology. \\( z = \square \\) (round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for the z - test statistic

The formula for the z - test statistic in a one - sample z - test is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\), where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean under the null hypothesis, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Step2: Identify the values of \(\bar{x}\), \(\mu\), \(\sigma\), and \(n\)

We are given that \(\bar{x} = 750\) hours, \(\mu=774\) hours (from \(H_0:\mu\geq774\)), \(\sigma = 65\) hours, and \(n = 27\).

Step3: Substitute the values into the formula

$$
LATEXBLOCK0
$$

First, calculate \(\sqrt{27}\approx5.196\), then \(\frac{65}{\sqrt{27}}\approx\frac{65}{5.196}\approx12.51\)

$$ z=\frac{-24}{12.51}\approx - 1.92 $$

Answer:

\(-1.92\)