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: $\bar{x}_1 = 1235$, $n_1 = 35$, $\bar{x}_2 = 1195$, and $n_2 = 55$. population parameters: $\sigma_1 = 75$ and $\sigma_2 = 100$. (a) the test statistic for $\mu_1 - \mu_2$ is 40. (b) the standardized test statistic for $\mu_1 - \mu_2$ is 2.16. (round to two decimal places as needed.) (c) is the standardized test statistic in the rejection region? \bigcirc no \bigcirc yes

Explanation:

Step1: Determine the critical value

For a left - tailed test with \(\alpha = 0.01\) and two - population z - test (since population standard deviations are known), the critical value \(z_{\alpha}\) is found from the standard normal distribution. The critical value for \(\alpha=0.01\) (left - tailed) is \(z=- 2.33\) (because \(P(Z < - 2.33)\approx0.01\)).

Step2: Compare the standardized test statistic with the critical value

The standardized test statistic \(z = 2.16\) (from part (b)). For a left - tailed test with \(\mu_1<\mu_2\), the rejection region is \(z < - z_{\alpha}\), i.e., \(z < - 2.33\). Since \(2.16>-2.33\), the standardized test statistic is not in the rejection region.

Answer:

No