Find the standardized test statistic, t, to test the claim that $\mu _ { 1 }

Question

Find the standardized test statistic, t, to test the claim that $\mu _ { 1 } < \mu _ { 2 } .$ Two samples are random, independent, and come from populations that are normally distributed. The sample statistics are given below. Assume that $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$ $$\begin{array} { l l } 
\mathrm { n } _ { 1 } = 15 & \mathrm { n } _ { 2 } = 15 \
\overline { \mathrm { x } } 1 = 21.2 & \overline { \mathrm { x } } 2 = 23.75 \
\mathrm {~s} _ { 1 } = 2.9 & \mathrm {~s} _ { 2 } = 2.8
\end{array}$$
A)-2.450
B)-3.165
C)-1.667
D)-0.669

Quizplus · Accepted Answer

The correct choice is A because the standardized test statistic, t, is calculated as:$$t = \frac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$Since $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$, the formula simplifies to:$$t = \frac{(\overline{x}_1 - \overline{x}_2)}{\sqrt{\frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2}}}$$Plugging in the given values, we get:$$t = \frac{(21.2 - 23.75)}{\sqrt{\frac{2.9^2}{15} + \frac{2.8^2}{15}}} = -2.450$$Therefore, the standardized test statistic, t, is -2.450.

Find the Standardized Test Statistic, T, to Test the Claim μ1<μ2.\mu _ { 1 } < \mu _ { 2 } .μ1​<μ2​.

Find the Standardized Test Statistic, T, to Test the Claim $\mu _ { 1 } < \mu _ { 2 } .$