Question

the mean exam score for 45 male high school students is 24.1 and the population standard deviation is 4.5. the mean exam score for 52 female high school students is 21.1 and the population standard deviation is 4.2. at \\( \alpha = 0.01 \\), can you reject the claim that male and female high school students have equal exam scores? complete parts (a) through (e).
click here to view page 1 of the standard normal distribution table.
click here to view page 2 of the standard normal distribution table.
answers as needed.)
what is/are the rejection region(s)?
\\( \bigcirc \\) a. \\( z < - 1.64, z > 1.64 \\) \\( \bigcirc \\) b. \\( z < - 2.33, z > - 2.33 \\)
\\( \bigcirc \\) c. \\( z > 2.58 \\) \\( \bigcirc \\) d. \\( z < 1.64 \\)
\\( \bigcirc \\) e. \\( z < - 2.33 \\) \\( \bigcirc \\) f. \\( z > - 3.08 \\)
\\( \bigcirc \\) g. \\( z < - 2.58, z > 2.58 \\) \\( \bigcirc \\) h. \\( z < - 3.08, z > - 3.08 \\)
(c) find the standardized test statistic \\( z \\) for \\( \mu _ { 1 } - \mu _ { 2 } \\).
\\( z = 3.37 \\) (round to two decimal places as needed.)
(d) decide whether to reject or fail to reject the null hypothesis. choose the correct answer below.
\\( \bigcirc \\) a. reject \\( h _ { 0 } \\). the standardized test statistic is not in the rejection region.
\\( \bigcirc \\) b. fail to reject \\( h _ { 0 } \\). the standardized test statistic is in the rejection region.
\\( \bigcirc \\) c. fail to reject \\( h _ { 0 } \\). the standardized test statistic is not in the rejection region.
\\( \bigcirc \\) d. reject \\( h _ { 0 } \\). the standardized test statistic is in the rejection region.

Explanation:

Step1: Analyze the rejection region

The claim is that male and female high - school students have equal exam scores. So the null hypothesis \(H_0:\mu_1=\mu_2\) and the alternative hypothesis \(H_1:\mu_1
eq\mu_2\) (a two - tailed test). Given \(\alpha = 0.01\), for a two - tailed z - test, the critical values are \(z=\pm z_{\alpha/2}\). Since \(\alpha=0.01\), then \(\alpha/2 = 0.005\). Looking up in the standard normal distribution table, \(z_{0.005}=2.58\). The rejection regions are \(z < - 2.58\) or \(z>2.58\).

Step2: Compare the test statistic with the rejection region

The standardized test statistic \(z = 3.37\). Since \(3.37>2.58\), the test statistic is in the rejection region.

Answer:

D. Reject \(H_0\). The standardized test statistic is in the rejection region.