a professor believes that, for the introducto...
a professor believes that, for the introductory art history classes at his university, the mean test score of students in the evening classes is lower than the mean test score of students in the morning classes. he collects data from a random sample of 250 students in evening classes and finds that they have a mean test score of 83.8. he knows the population standard deviation for the evening classes to be 2.6 points. a random sample of 150 students from morning classes results in a mean test score of 84.2. he knows the population standard deviation for the morning classes to be 1.9 points. test his claim with a 90% level of confidence. let students in the evening classes be population 1 and let students in the morning classes be population 2.\n\nstep 2 of 3: compute the value of the test statistic. round your answer to two decimal places.
Answer
# Explanation:
## Step1: Identify the formula for two - sample z - test statistic
The formula for the two - sample z - test statistic when population standard deviations $\sigma_1$ and $\sigma_2$ are known is $z=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}}$. Here, the null hypothesis $H_0:\mu_1\geq\mu_2$ and the alternative hypothesis $H_1:\mu_1 < \mu_2$, so under $H_0$, $(\mu_1-\mu_2) = 0$.
## Step2: Define the given values
We have $\bar{x}_1 = 83.8$ (sample mean of evening classes), $n_1=250$ (sample size of evening classes), $\sigma_1 = 2.6$ (population standard deviation of evening classes), $\bar{x}_2=84.2$ (sample mean of morning classes), $n_2 = 150$ (sample size of morning classes), $\sigma_2=1.9$ (population standard deviation of morning classes), and $(\mu_1 - \mu_2)=0$ under $H_0$.
## Step3: Substitute the values into the formula
\[
\begin{align*}
z&=\frac{(83.8 - 84.2)-0}{\sqrt{\frac{2.6^{2}}{250}+\frac{1.9^{2}}{150}}}\\
&=\frac{- 0.4}{\sqrt{\frac{6.76}{250}+\frac{3.61}{150}}}\\
&=\frac{-0.4}{\sqrt{0.02704 + 0.02407}}\\
&=\frac{-0.4}{\sqrt{0.05111}}\\
&=\frac{-0.4}{0.2261}\\
&\approx - 1.77
\end{align*}
\]
# Answer:
$-1.77$