A test was made of $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ versus $H _ { 1 } : \mu _ { 1 }

Question

A test was made of $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ versus $H _ { 1 } : \mu _ { 1 } < \mu _ { 2 }$. The sample means were $\bar { x } _ { 1 } = 8$ and $\bar { x } _ { 2 } = 14$, the sample standard deviations were $s _ { 1 } = 6$ and $s _ { 2 } = 8$, and the sample sizes were $n _ { 1 } = 12$ and $n _ { 2 } = 11$. Is $H _ { 0 }$ rejected at the 0.05 level? (Hint: First compute the value of the test statistic.)

Quizplus · Accepted Answer

The test statistic for this problem is given by:

$$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$

where $s_p$ is the pooled standard deviation, which is calculated as:

$$s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$$

Plugging in the values from the problem, we get:

$$s_p = \sqrt{\frac{(12 - 1)(6)^2 + (11 - 1)(8)^2}{12 + 11 - 2}} = 7.02$$

$$t = \frac{8 - 14}{7.02\sqrt{\frac{1}{12} + \frac{1}{11}}} = -2.55$$

Using a t-table with 21 degrees of freedom (rounded from the pooled degrees of freedom of 21.98), we find that the critical value for a one-tailed test at the 0.05 level is -1.721. Since our calculated t-value of -2.55 is smaller in magnitude than the critical value, we reject the null hypothesis at the 0.05 level.

A Test Was Made Of H0:μ1=μ2H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }H0​:μ1​=μ2​

A Test Was Made Of $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$