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)

Explanation:

Step1: Understand the claim

The manufacturer's claim is that the mean life of a certain type of light bulb is at least 765 hours. So the claim is \(\mu\geq765\).

Step2: Define null and alternative hypotheses

The null hypothesis \(H_0\) is the statement being tested. The alternative hypothesis \(H_a\) is the statement we are trying to find evidence for.
If the claim is \(\mu\geq765\), then the null hypothesis \(H_0:\mu\geq765\) (claim) and the alternative hypothesis \(H_a:\mu < 765\)

Answer:

E. \(H_0:\mu\geq765\) (claim), \(H_a:\mu < 765\)