Question

a fast food restaurant estimates that the mean sodium content in one of its breakfast sandwiches is no more than 912 milligrams. a random sample of 30 breakfast sandwiches has a mean sodium content of 901 milligrams. assume the population standard deviation is 18 milligrams. at \\( \alpha = 0.10 \\), do you have enough evidence to reject the restaurants claim? complete parts (a) through (e).
(a) identify the null hypothesis and alternative hypothesis.
a. \\( h _ { 0 } : \mu \leq 901 \\)
\\( h _ { a } : \mu < 901 \\) (claim)
b. \\( h _ { 0 } : \mu \
eq 912 \\) (claim)
\\( h _ { a } : \mu = 912 \\)
c. \\( h _ { 0 } : \mu > 912 \\)
\\( h _ { a } : \mu \leq 912 \\) (claim)
d. \\( h _ { 0 } : \mu = 901 \\) (claim)
\\( h _ { a } : \mu \
eq 901 \\)
e. \\( h _ { 0 } : \mu \leq 912 \\) (claim)
\\( h _ { a } : \mu > 912 \\)
f. \\( h _ { 0 } : \mu < 901 \\) (claim)
\\( h _ { a } : \mu \geq 901 \\)
(b) identify the critical value(s). use technology.
\\( z _ { 0 } = 1.28 \\)
(use a comma to separate answers as needed. round to two decimal places as needed.)
identify the rejection region(s). select the correct choice below.
a. the rejection region is \\( z > 1.28 \\).
b. the rejection regions are \\( z > 1.28 \\) and \\( z < - 1.28 \\).
c. the rejection region is \\( z < 1.28 \\).

Explanation:

Step1: Recall hypothesis testing concepts

The restaurant claims that the mean sodium content $\mu\leq912$. So the null hypothesis $H_0:\mu\leq912$ (claim) and the alternative hypothesis $H_a:\mu > 912$. This is a right - tailed test.

Step2: Determine rejection region for right - tailed test

For a right - tailed test with significance level $\alpha = 0.10$, the critical value $z_0$ is such that $P(Z>z_0)=\alpha$. Using standard normal distribution tables or technology, we find $z_0 = 1.28$. In a right - tailed test, the rejection region is where the test statistic is greater than the critical value.

Answer:

A. The rejection region is $z>1.28$.