Question

conduct a test at the $alpha = 0.10$ level of significance by determining (a) the null and alternative hypotheses, (b) the test statistic, and (c) the p - value. assume the samples were obtained independently from a large population using simple random sampling. test whether $p_1>p_2$. the sample data are $x_1 = 126$, $n_1 = 257$, $x_2 = 135$, and $n_2 = 317$. (a) choose the correct null and alternative hypotheses below. ○ a. $h_0:p_1 = p_2$ versus $h_1:p_1>p_2$ ○ b. $h_0:p_1 = p_2$ versus $h_1:p_1p_2$ ○ d. $h_0:p_1 = p_2$ versus $h_1:p_1
eq p_2$

Explanation:

Step1: Understand Hypothesis Testing Basics

In hypothesis testing for proportions \(p_1\) and \(p_2\), the null hypothesis \(H_0\) is a statement of equality. The alternative hypothesis \(H_1\) is based on the claim. Here, the claim is \(p_1>p_2\).

Step2: Analyze Each Option

• Option A: \(H_0:p_1 = p_2\) (null hypothesis of equality) and \(H_1:p_1>p_2\) (matches the claim).
• Option B: \(H_1:p_1 < p_2\) does not match the claim \(p_1>p_2\).
• Option C: \(H_0:p_1 = 0\) is not a standard null hypothesis for comparing two proportions \(p_1\) and \(p_2\).
• Option D: \(H_1:p_1

eq p_2\) is a two - tailed test, while our claim \(p_1>p_2\) is one - tailed.

Answer:

A. \(H_0:p_1 = p_2\) versus \(H_1:p_1>p_2\)