Question

for the given data, (a) find the test statistic, (b) find the standardized test statistic, (c) decide whether the standardized test statistic is in the rejection region, and (d) decide whether you should reject or fail to reject the null hypothesis. the samples are random and independent. claim: \\( \mu _ { 1 } < \mu _ { 2 }, \alpha = 0.01 \\). sample statistics: \\( \overline { x } _ { 1 } = 1215, n _ { 1 } = 35, \overline { x } _ { 2 } = 1185 \\), and \\( n _ { 2 } = 65 \\). population parameters: \\( \sigma _ { 1 } = 80 \\) and \\( \sigma _ { 2 } = 115 \\). (a) the test statistic for \\( \mu _ { 1 } - \mu _ { 2 } \\) is 30. (b) the standardized test statistic for \\( \mu _ { 1 } - \mu _ { 2 } \\) is 1.53. (round to two decimal places as needed.) (c) is the standardized test statistic in the rejection region? no yes

Explanation:

Step1: Determine the rejection region

Since the claim is \( \mu_1<\mu_2\), the null hypothesis is \(H_0:\mu_1 - \mu_2\geq0\) and the alternative hypothesis is \(H_a:\mu_1-\mu_2 < 0\). This is a left - tailed test. For \(\alpha = 0.01\), the critical value \(z_{\alpha}\) is \(z_{0.01}=- 2.33\). The rejection region is \(z < - 2.33\).

Step2: Compare the standardized test statistic with the critical value

The standardized test statistic \(z = 1.53\) (from part (b)).

Answer:

c. No