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).
(a) identify the null hypothesis and alternative hypothesis.
a. \\( h _ { 0 } : \mu \leq 750 \\)
\\( h _ { a } : \mu > 750 \\) (claim)
b. \\( h _ { 0 } : \mu > 774 \\)
\\( h _ { a } : \mu \leq 774 \\) (claim)
c. \\( h _ { 0 } : \mu = 750 \\)
\\( h _ { a } : \mu \
eq 750 \\) (claim)
d. \\( h _ { 0 } : \mu \geq 774 \\) (claim)
\\( h _ { a } : \mu < 774 \\)
e. \\( h _ { 0 } : \mu < 750 \\) (claim)
\\( h _ { a } : \mu \geq 750 \\)
f. \\( h _ { 0 } : \mu \
eq 774 \\) (claim)
\\( h _ { a } : \mu = 774 \\)
(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 type of test

Since the alternative hypothesis is \(H_{a}:\mu < 774\), this is a left - tailed test.

Step2: Find the critical value

For a left - tailed test with \(\alpha = 0.08\), we use the standard normal distribution \(Z\). The critical value \(z_{0}\) is the \(z\) - score such that \(P(Z<z_{0})=\alpha = 0.08\).
Using technology (e.g., a calculator with a normal distribution function or statistical software), we find \(z_{0}\approx - 1.41\)

Answer:

\(-1.41\)