Test Hypotheses Regarding the Difference of Two Independent Means
-Find the standardized test statistic to test the hypothesis that $\mu _ { 1 }

Question

Test Hypotheses Regarding the Difference of Two Independent Means
-Find the standardized test statistic to test the hypothesis that $\mu _ { 1 } < \mu _ { 2 }$. Two samples are randomly selected from each population. The sample statistics are given below. Use $\alpha = 0.05$.
$$\begin{array} { l l } 
\mathrm { n } _ { 1 } = 35 & \mathrm { n } _ { 2 } = 42 \
\overline { \mathrm { x } } 1 = 20.04 & \overline { \mathrm { x } } 2 = 22.59 \
\mathrm {~s} _ { 1 } = 2.9 & \mathrm {~s} _ { 2 } = 2.8
\end{array}$$
A) $- 3.90$
B) $- 3.16$
C) $- 2.63$
D) $- 1.66$

Quizplus · Accepted Answer

The standardized test statistic is calculated as:$$z = \frac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$Since we are testing the hypothesis that $\mu_1 < \mu_2$, we have $\mu_1 - \mu_2 < 0$. Substituting the given values into the formula, we get:$$z = \frac{(20.04 - 22.59) - 0}{\sqrt{\frac{2.9^2}{35} + \frac{2.8^2}{42}}} = -3.90$$Therefore, the standardized test statistic is -3.90, which corresponds to choice A.

Test Hypotheses Regarding the Difference of Two Independent Means -Find μ1<μ2\mu _ { 1 } < \mu _ { 2 }μ1​<μ2​

Test Hypotheses Regarding the Difference of Two Independent Means
-Find $\mu _ { 1 } < \mu _ { 2 }$