Question

suppose you want to test the claim that $\mu_1 < \mu_2$. two samples are randomly selected from statistics are given below. at a level of significance of $\alpha = 0.05$, when should you reject $n_1 = 35, n_2 = 42, \bar{x}_1 = 21.72, \bar{x}_2 = 24.27, \sigma_1 = 2.9, \sigma_2 = 2.8$ \bigcirc a. reject $h_0$ if the standardized test statistic is less than $-2.575$. \bigcirc b. reject $h_0$ if the standardized test statistic is less than $-1.645$. \bigcirc c. reject $h_0$ if the standardized test statistic is less than $-1.96$. \bigcirc d. reject $h_0$ if the standardized test statistic is less than $-2.33$.

Explanation:

Step1: Identify Test Type

This is a one - tailed (left - tailed) z - test for two population means, since we are testing \( \mu_1<\mu_2 \) and population standard deviations \( \sigma_1,\sigma_2 \) are known. The null hypothesis \( H_0:\mu_1=\mu_2 \), alternative hypothesis \( H_a:\mu_1 < \mu_2 \).

Step2: Determine Critical Value

For a left - tailed test with \( \alpha = 0.05 \), we look up the z - critical value. The z - critical value for a left - tailed test at \( \alpha=0.05 \) is \( z_{\alpha}=- 1.645 \). This is because the area to the left of \( z=- 1.645 \) under the standard normal curve is 0.05. So we reject \( H_0 \) when the standardized test statistic is less than \( - 1.645 \).

Answer:

B. Reject \( H_0 \) if the standardized test statistic is less than \( - 1.645 \)