Question

test the claim about the difference between two population means \\( \mu_1 \\) and \\( \mu_2 \\) at the level of signili samples are random and independent, and the populations are normally distributed.
claim: \\( \mu_1 \leq \mu_2 \\); \\( \alpha = 0.10 \\). assume \\( \sigma_1^2 \
eq \sigma_2^2 \\)
sample statistics: \\( \bar{x}_1 = 2402 \\), \\( s_1 = 170 \\), \\( n_1 = 12 \\) and
\\( \bar{x}_2 = 2296 \\), \\( s_2 = 51 \\), \\( n_2 = 10

identify the null and alternative hypotheses. choose the correct answer below.
\\( \bigcirc \\) a. \\( h_0: \mu_1 < \mu_2 \\)
\\( h_a: \mu_1 \geq \mu_2 \\)
\\( \bigcirc \\) b. \\( h_0: \mu_1 \leq \mu_2 \\)
\\( h_a: \mu_1 > \mu_2 \\)
\\( \bigcirc \\) c. \\( h_0: \mu_1 > \mu_2 \\)
\\( h_a: \mu_1 \leq \mu_2 \\)
\\( \bigcirc \\) d. \\( h_0: \mu_1 \
eq \mu_2 \\)
\\( h_a: \mu_1 = \mu_2 \\)
\\( \bigcirc \\) e. \\( h_0: \mu_1 = \mu_2 \\)
\\( h_a: \mu_1 \
eq \mu_2 \\)
\\( \bigcirc \\) f. \\( h_0: \mu_1 \geq \mu_2 \\)
\\( h_a: \mu_1 < \mu_2 \\)

Explanation:

Step1: Recall Hypothesis Testing Rules

In hypothesis testing, the null hypothesis (\(H_0\)) contains the equality or the claim we are testing (or its complement in some cases), and the alternative hypothesis (\(H_a\)) is the opposite of the null, representing what we might believe if we reject \(H_0\). The claim here is \(\mu_1 \leq \mu_2\). So the null hypothesis should include this claim (or the equality part if the claim has an inequality), and the alternative hypothesis is the opposite.

Step2: Analyze Each Option

• Option A: \(H_0: \mu_1 < \mu_2\) (null has strict inequality, which is not standard; null should have equality or the claim's inequality with equality included). Incorrect.
• Option B: \(H_0: \mu_1 \leq \mu_2\) (matches the claim as the null hypothesis, since we test the claim in the null or against it) and \(H_a: \mu_1 > \mu_2\) (opposite of the null, which is correct because if we reject \(H_0: \mu_1 \leq \mu_2\), we conclude \(\mu_1 > \mu_2\)).
• Option C: \(H_0: \mu_1 > \mu_2\) (null is opposite of the claim), incorrect.
• Option D: \(H_0: \mu_1

eq \mu_2\) (null is not related to the claim), incorrect.

• Option E: \(H_0: \mu_1 = \mu_2\) (null is equality, not matching the claim's inequality), incorrect.
• Option F: \(H_0: \mu_1 \geq \mu_2\) (null is opposite of the claim), incorrect.

Answer:

B. \(H_0: \mu_1 \leq \mu_2\), \(H_a: \mu_1 > \mu_2\)