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.
\\( \bigcirc \\) a. \\( h_{0}: \mu \leq 901 \\)
\\( h_{a}: \mu<901 \\) (claim)
\\( \bigcirc \\) b. \\( h_{0}: \mu \
eq 912 \\) (claim)
\\( h_{a}: \mu=912 \\)
\\( \bigcirc \\) c. \\( h_{0}: \mu>912 \\)
\\( h_{a}: \mu \leq 912 \\) (claim)
\\( \bigcirc \\) d. \\( h_{0}: \mu=901 \\) (claim)
\\( h_{a}: \mu \
eq 901 \\)
\\( \bigcirc \\) e. \\( h_{0}: \mu \leq 912 \\) (claim)
\\( h_{a}: \mu>912 \\)
\\( \bigcirc \\) f. \\( h_{0}: \mu<901 \\) (claim)
\\( h_{a}: \mu \geq 901 \\)
(b) identify the critical value(s) use technology.
\\( z_{0}=\square \\)
(use a comma to separate answers as needed. round to two decimal places as needed.)

Explanation:

Step1: Identify the hypotheses

The restaurant claims that the mean sodium content is no more than \(912\) milligrams. So the null hypothesis \(H_{0}\) is \(\mu\leq912\) (the claim) and the alternative hypothesis \(H_{a}\) is \(\mu > 912\). This is a right - tailed test.

Step2: Find the critical value

For a right - tailed test with \(\alpha=0.10\), we look up the \(z\) - value in the standard normal distribution table. The critical value \(z_{0}\) is the value such that \(P(Z>z_{0})=\alpha = 0.10\), or \(P(Z\leq z_{0})=1 - \alpha=0.90\). Using a standard normal table or technology (e.g., a TI - 84 Plus: invNorm(0.90,0,1)), we find \(z_{0}=1.28\)

Answer:

\(z_{0} = 1.28\)