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 \(H_{a}: \mu_{1} \neq \mu_{2}\) using the sample results \(\bar{x}_{1}=15.3, s_{1}=11.6\) with \(n_{1}=100\) and \(\bar{x}_{2}=18.4, s_{2}=14.3\) with \(n_{2}=80\).

Short answer

After performing all these steps, the short answer will be whether we reject or fail to reject the null hypothesis, depending on the comparison between the calculated t statistic and the critical value.

Step-by-step solution

Identify the Null and Alternative Hypotheses

The null hypothesis \(H_{0}: \mu_{1}=\mu_{2}\) states that the population means are equal, whereas the alternative hypothesis \(H_{a}: \mu_{1} \neq \mu_{2}\) states that they are not equal.

Calculate the Pooled Standard Deviation

The pooled standard deviation for independent samples is given by the formula \[\sqrt{\frac{(n_1 - 1)s_1^2 +(n_2 - 1)s_2^2 }{n_1 + n_2 - 2}}\] Applying the given values from the problem \(\sqrt{\frac{(100 - 1)11.6^2 +(80 - 1)14.3^2 }{100 + 80 - 2}}\) we find the pooled standard deviation.

Calculate the t Statistic

The t statistic is given by the formula \[\frac{(\bar{x}_{1}-\bar{x}_{2}) - ( \mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^2}{n_{1}}+\frac{s_{2}^2}{n_{2}}}}\] Since we are testing for equal population means, \(\mu_{1}-\mu_{2}=0\). Thus, the t statistic becomes \[\frac{(15.3-18.4)}{\sqrt{\frac{11.6^2}{100}+\frac{14.3^2}{80}}}\] Calculate the value of t statistic.

Find the Critical Value

The critical value is found in the t distribution table. The degrees of freedom is given by \(n_{1} + n_{2} - 2= 100 + 80 - 2 = 178\). Assuming a 5% level of significance for a two-tailed test, the critical value can be found in the corresponding row of the t-distribution table.

Compare t Statistic and Critical Value

Finally, we compare the calculated t statistic with the critical value. If the absolute value of t statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.