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).

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 \\).

(c) identify the standardized test statistic. use technology.

\\( z = - 3.35 \\) (round to two decimal places as needed.)

(d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

a. reject \\( h _ { 0 } \\). there is sufficient evidence to reject the claim that mean sodium content is no more than 912 milligrams.
b. fall to reject \\( h _ { 0 } \\). there is sufficient evidence to reject the claim that mean sodium content is no more than 912 milligrams.
c. reject \\( h _ { 0 } \\). there is not sufficient evidence to reject the claim that mean sodium content is no more than 912 milligrams.
d. fail to reject \\( h _ { 0 } \\). there is not sufficient evidence to reject the claim that mean sodium content is no more than 912 milligrams.

Explanation:

Step1: Analyze the rejection region

The claim is \( \mu\leq912 \), so the null hypothesis \( H_0:\mu\leq912 \) and the alternative hypothesis \( H_1:\mu > 912 \) (right - tailed test). For a right - tailed test with \( \alpha = 0.10 \), the critical value \( z_{\alpha}\) is such that \( P(Z>z_{\alpha})=\alpha \). Using the standard normal distribution table, \( z_{0.10}=1.28 \). The rejection region is \( z > 1.28 \).

Step2: Compare the test statistic with the rejection region

The standardized test statistic \( z=- 3.35 \). Since \( -3.35<1.28 \), the test statistic does not fall in the rejection region \( z > 1.28 \).

Answer:

D. Fail to reject \( H_0 \). There is not sufficient evidence to reject the claim that mean sodium content is no more than 912 milligrams.