Question

a light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 761 hours. a random sample of 24 light bulbs has a mean life of 733 hours. assume the population is normally distributed and the population standard deviation is 62 hours. at α = 0.02, do you have enough evidence to reject the manufacturers claim? complete parts (a) through (e)
(a) identify the null hypothesis and alternative hypothesis.
a. ( h_{0}: mugeq761 ) (claim)
( h_{a}: mu < 761 )
b. ( h_{0}: mu = 733 )
( h_{a}: mu
eq733 ) (claim)
c. ( h_{0}: muleq733 ) (claim)
( h_{a}: mugeq733 )
d. ( h_{0}: muleq733 )
( h_{a}: mu > 733 ) (claim)
e. ( h_{0}: mu > 761 )
( h_{a}: muleq761 ) (claim)
f. ( h_{0}: mu
eq761 ) (claim)
( h_{a}: mu = 761 )
(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
(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 \(n\)

From the problem, \(\bar{x} = 733\) hours, \(\mu=761\) hours (from the null hypothesis \(H_0:\mu\geq761\)), \(\sigma = 62\) hours, and \(n = 24\).

Step3: Calculate the z - test statistic

Substitute the values into the formula:

$$
LATEXBLOCK0
$$

Answer:

\(z\approx - 2.21\)