Question

Use the t-distribution and the given sample results to complete the test of the given hypotheses. Assume the results come from random samples, and if the sample sizes are small, assume the underlying distributions are relatively normal. Test \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1}>\mu_{2}\) using the sample results \(\bar{x}_{1}=56, s_{1}=8.2\) with \(n_{1}=30\) and \(\bar{x}_{2}=51, s_{2}=6.9\) with \(n_{2}=40\).

Short answer

To solve this problem, calculate the standard error and the t-test statistic using the given sample means, standard deviations and sizes. Then, compare the test statistic to the critical value that corresponds to 0.05 significance level and the calculated degrees of freedom. If the test statistic exceeds the critical value, reject the null hypothesis, indicating that the mean of group 1 is larger than the mean of group 2.

Step-by-step solution

Calculate the standard error

The standard error (SE) of the difference in the means, denoted as \( SE(\bar{x}_{1}-\bar{x}_{2})\), is calculated as follows: \[SE(\bar{x}_{1}-\bar{x}_{2}) = \sqrt{\left(\frac{s_{1}^2}{n_{1}}\right) + \left(\frac{s_{2}^2}{n_{2}}\right)}\] Substituting the given values, we get \[SE(\bar{x}_{1}-\bar{x}_{2}) = \sqrt{\left(\frac{8.2^2}{30}\right) + \left(\frac{6.9^2}{40}\right)}.\]

Calculate the test statistic

The test statistic (t) is given by the formula: \[t = \frac{(\bar{x}_{1} - \bar{x}_{2}) - (μ_{1} - μ_{2})}{SE(\bar{x}_{1}-\bar{x}_{2})}\]. Now, under null hypothesis, \( μ_{1} = μ_{2} \), so the equation simplifies to \[t = \frac{(\bar{x}_{1}-\bar{x}_{2})}{SE(\bar{x}_{1}-\bar{x}_{2})}\]. Substituting the given values, we find the test statistic.

Compare to critical value

The critical value for the t-statistic depends on the significance level (normally 0.05) and the degrees of freedom. The degrees of freedom can be approximated as the smaller of \(n_{1}-1\) and \(n_{2}-1\), yielding the smaller of 29 and 39, which is 29. With significance level 0.05 for a one-tailed test and 29 degrees of freedom, the critical value can be obtained from a t-distribution table. If the calculated t-statistic is higher than the critical value, we reject the null hypothesis.