Question

a light bulb manufacturer guarantees that the mean life of a certain type of light bulb is at least 765 hours. a random sample of 23 light bulbs has a mean life of 749 hours. assume the population is normally distributed and the population standard deviation is 58 hours. at \\( \alpha = 0.05 \\), do you have enough evidence to reject the manufacturers claim? complete parts (a) through (e).
(a) identify the null hypothesis and alternative hypothesis.
\\( \bigcirc \\) a. \\( h_{0}: \mu \
eq 765 \\) (claim) \\( h_{a}: \mu = 765 \\)
\\( \bigcirc \\) b. \\( h_{0}: \mu>765 \\) \\( h_{a}: \mu \leq 765 \\) (claim)
\\( \bigcirc \\) c. \\( h_{0}: \mu \leq 749 \\) \\( h_{a}: \mu>749 \\) (claim)
\\( \bigcirc \\) d. \\( h_{0}: \mu<749 \\) (claim) \\( h_{a}: \mu \geq 749 \\)
\\( \bigcirc \\) e. \\( h_{0}: \mu \geq 765 \\) (claim) \\( h_{a}: \mu<765 \\)
\\( \bigcirc \\) f. \\( h_{0}: \mu = 749 \\) \\( h_{a}: \mu \
eq 749 \\) (claim)
(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: Identify the type of test

Since the claim is $\mu\geq765$ and we are testing against $\mu < 765$, this is a left - tailed test.

Step2: Find the critical value

For a left - tailed test with $\alpha = 0.05$, we look up the z - value in the standard normal distribution table. The critical value \(z_0\) is the value such that \(P(Z<z_0)=\alpha\). Using a standard normal table or technology, for \(\alpha = 0.05\), \(z_0=- 1.645\approx - 1.65\)

Answer:

\(-1.65\)