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 \\(\square\\).

Explanation:

Step1: Recall the formula for the test statistic (difference in sample means)

The test statistic for the difference between two population means \(\mu_1 - \mu_2\) when the population standard deviations \(\sigma_1\) and \(\sigma_2\) are known is given by the difference in the sample means, i.e., \(\bar{x}_1 - \bar{x}_2\). Wait, no, actually, for part (a), the test statistic for \(\mu_1 - \mu_2\) in the context of a two - sample z - test (since population standard deviations are known) is the difference in the sample means. Wait, let's clarify:

The formula for the test statistic (the difference in sample means) is \(\bar{x}_1-\bar{x}_2\). But also, the standardized test statistic (z - statistic) is given by:

$$z=\frac{(\bar{x}_1 - \bar{x}_2)-(\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}}$$

But for part (a), the test statistic for \(\mu_1-\mu_2\) is \(\bar{x}_1 - \bar{x}_2\) (the difference in the sample means).

We are given \(\bar{x}_1 = 1235\) and \(\bar{x}_2=1195\).

Step2: Calculate the test statistic

Substitute the values of \(\bar{x}_1\) and \(\bar{x}_2\) into the formula for the test statistic (difference in sample means):

\(\bar{x}_1-\bar{x}_2=1235 - 1195=40\)

Answer:

\(40\)