Find the standardized test statistic, t , to test the hypothesis that $\mu _ { 1 } = \mu _ { 2 }$ Two samples are randomly selected and come from populations that are normal. The sample statistics are given below.
$\begin{array} { l l }
\mathrm { n } _ { 1 } = 25 & \mathrm { n } _ { 2 } = 30 \
\overline { \mathrm { x } } 1 = 16 & \overline { \mathrm { x } } 2 = 14 \
\mathrm {~s} _ { 1 } = 1.5 & \mathrm {~s} _ { 2 } = 1.9
\end{array}$
A) 4.361
B) 3.287
C) 2.892
D) 1.986

Question

Find the standardized test statistic,  t , to test the hypothesis that $\mu _ { 1 } = \mu _ { 2 }$ Two samples are randomly selected and come from populations that are normal. The sample statistics are given below. 
$\begin{array} { l l } 
\mathrm { n } _ { 1 } = 25 & \mathrm { n } _ { 2 } = 30 \
\overline { \mathrm { x } } 1 = 16 & \overline { \mathrm { x } } 2 = 14 \
\mathrm {~s} _ { 1 } = 1.5 & \mathrm {~s} _ { 2 } = 1.9
\end{array}$
A)  4.361 
B)  3.287 
C)  2.892 
D)  1.986

Quizplus · Accepted Answer

The correct answer is A because the standardized test statistic 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 we are testing the hypothesis that $\mu _ { 1 } = \mu _ { 2 }$, the numerator simplifies to:$$(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2) = \overline{x}_1 - \overline{x}_2$$Substituting the given values, we get:$$t = \frac{(16 - 14) - (0)}{\sqrt{\frac{1.5^2}{25} + \frac{1.9^2}{30}}} = 4.361$$

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

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