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}: mu geq 761 ) (claim)
( h_{a}: mu<761 )
b. ( h_{0}: mu = 733 )
( h_{a}: mu
eq 733 ) (claim)
c. ( h_{0}: mu<733 ) (claim)
( h_{a}: mu geq 733 )
d. ( h_{0}: mu leq 733 )
( h_{a}: mu>733 ) (claim)
e. ( h_{0}: mu>761 )
( h_{a}: mu leq 761 ) (claim)
f. ( h_{0}: mu
eq 761 ) (claim)
( h_{a}: mu = 761 )
(b) identify the critical value(s). use technology
( z_{0}= ) (use a comma to separate answers as needed. round to two decimal places as needed.)

Explanation:

Step1: Determine the null and alternative hypotheses

The manufacturer claims that the mean life of a certain type of light bulb is at least \(761\) hours. So the null hypothesis \(H_{0}\) is the claim, \(H_{0}:\mu\geq761\) (claim), and the alternative hypothesis \(H_{a}:\mu < 761\)

Step2: Find the critical value

Since the test is a left - tailed test (\(H_{a}:\mu < 761\)) and \(\alpha = 0.02\), we use the standard normal distribution \(Z\).
Using technology (e.g., a TI - 84 Plus: invNorm(0.02,0,1)), the critical value \(z_{0}\) is \(z_{0}=- 2.05\)

Answer:

(a) \(H_{0}:\mu\geq761\) (claim), \(H_{a}:\mu < 761\)
(b) \(z_{0}=-2.05\)