Question

a light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 757 hours. a random sample of 20 light bulbs has a mean life of 728 hours. assume the population is normally distributed and the population standard deviation is 61 hours. at \\( \alpha = 0.02 \\), do you have enough evidence to reject the manufacturers claim? complete parts (a) through (e).
\\( h _ { a } : \mu < 757 \\)
(claim)
(claim)
\\( \bigcirc \mathrm { d } \\). \\( h _ { 0 } : \mu < 728 \\)
\\( \bigcirc \mathrm { e } \\). \\( h _ { 0 } : \mu \
eq 757 \\) (claim) \\( \bigcirc \mathrm { f } \\). \\( h _ { 0 } : \mu = 728 \\)
(claim)
\\( h _ { a } : \mu = 757 \\)
\\( h _ { a } : \mu \
eq 728 \\)
(claim)
(b) identify the critical value(s). use technology.
\\( z _ { 0 } = - 2.05 \\)
(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.
\\( \bigcirc \mathrm { a } \\).
\\( \bigcirc \mathrm { b } \\).
\\( \bigcirc \mathrm { 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 when the population standard deviation \(\sigma\) is known 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 \(\bar{x} = 728\) hours, \(\mu=757\) hours (from the manufacturer's claim which we test against), \(\sigma = 61\) hours, and \(n = 20\).

Step3: Substitute the values into the formula

$$
LATEXBLOCK0
$$

Answer:

\(z\approx - 2.13\)