Question

use technology to help you test the claim about the population mean, (mu), at the given level of significance, (alpha), using the given sample statistics. assume the population is normally distributed.
claim: (muleq1160); (alpha = 0.08); (sigma=199.88). sample statistics: (overline{x}=1185.66), (n = 200)
identify the null and alternative hypotheses. choose the correct answer below.
a. (h_{0}:mugeq1160) (h_{a}:mu<1160)
b. (h_{0}:mu>1160) (h_{a}:muleq1160)
c. (h_{0}:mugeq1185.66) (h_{a}:mu<1185.66)
d. (h_{0}:muleq1185.66) (h_{a}:mu>1185.66)
e. (h_{0}:mu>1185.66) (h_{a}:muleq1185.66)
f. (h_{0}:muleq1160) (h_{a}:mu>1160)

Explanation:

Step1: Understand null and alternative hypotheses

The null hypothesis \(H_0\) is a statement of no difference or a claim that we assume to be true initially. The alternative hypothesis \(H_a\) is the claim we are trying to find evidence for.
The claim is \(\mu\leq1160\). In hypothesis testing, the null hypothesis is the opposite of the alternative hypothesis (in a way that one of them must be true). So if the claim is \(\mu\leq1160\), the alternative hypothesis \(H_a\) will be \(\mu > 1160\) (since we want to test against the claim). And the null hypothesis \(H_0\) is the complement of the alternative hypothesis.

Step2: Analyze each option

• Option A: \(H_0:\mu\geq1160\), \(H_a:\mu < 1160\) is wrong because the claim is \(\mu\leq1160\) and we are not testing in this direction.
• Option B: \(H_0:\mu>1160\), \(H_a:\mu\leq1160\) is wrong because the null hypothesis should be a statement that includes equality (in the form of \(\geq\), \(\leq\) or \(=\)) in most cases (for a one - sided test related to a claim of \(\leq\)).
• Option C: \(H_0:\mu\geq1185.66\), \(H_a:\mu < 1185.66\) is wrong because \(1185.66\) is the sample mean, not the value in the claim.
• Option D: \(H_0:\mu\leq1185.66\), \(H_a:\mu > 1185.66\) is wrong because \(1185.66\) is the sample mean, not the value in the claim.
• Option E: \(H_0:\mu>1185.66\), \(H_a:\mu\leq1185.66\) is wrong because \(1185.66\) is the sample mean, not the value in the claim.
• Option F: \(H_0:\mu\leq1160\), \(H_a:\mu > 1160\). The null hypothesis \(H_0\) is the claim (since in hypothesis testing, when the claim has an equality (\(\leq\) in this case), we take the null hypothesis as the claim). The alternative hypothesis \(H_a\) is the opposite of the null hypothesis for a one - sided test.

Answer:

F. \(H_0:\mu\leq1160\), \(H_a:\mu > 1160\)