A researcher collects data on the distance between college students' high schools and their colleges. He believes that the median distance is 200 miles. A random sample of 15 students yields the following data: $$2525303045455575851101351751958102170$$
At $= .05$, should the researcher's hypothesis be accepted or rejected?
You may use the following abbreviated table of critical values (two tailed, $= .05$ ) taken from Table J:
$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline \mathrm{n} & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 \
\hline \text { critical valuc } & 1 & 2 & 2 & 3 & 3 & 3 & 4 & 4 & 4 & 5 & 5 & 5 & 6 & 6 & 6 \
\hline
\end{array}$
A) Reject because the test value 2 is less than the critical value 3
B) Do not reject because the test value 4 is less than the critical value 13
C) Reject because the test value 13 is more than the critical value 3
D) Do not reject because the test value 13 is more than the critical value 3

Question

A researcher collects data on the distance between college students' high schools and their colleges. He believes that the median distance is 200 miles. A random sample of 15 students yields the following data: $$2525303045455575851101351751958102170$$
At $= .05$, should the researcher's hypothesis be accepted or rejected?
You may use the following abbreviated table of critical values (two tailed, $= .05$ ) taken from Table J: 
$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline \mathrm{n} & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 \
\hline \text { critical valuc } & 1 & 2 & 2 & 3 & 3 & 3 & 4 & 4 & 4 & 5 & 5 & 5 & 6 & 6 & 6 \
\hline
\end{array}$
A) Reject because the test value 2 is less than the critical value 3
B) Do not reject because the test value 4 is less than the critical value 13
C) Reject because the test value 13 is more than the critical value 3
D) Do not reject because the test value 13 is more than the critical value 3

Quizplus · Accepted Answer

To test the hypothesis, we use the one-sample median test. We need to find the test statistic and compare it to the critical value. The formula for the test statistic is 
$$k = \dfrac{|n_1 - n_2| - 0.5(\mathrm{n})}{\sqrt{0.25 (\mathrm{n})}}$$
where $n_1$ is the number of observations less than the hypothesized median and $n_2$ is the number of observations greater than the hypothesized median. For this test, we have 
$$n_1 = 7, \quad n_2 = 8, \quad \mathrm{n} = 15.$$
Plugging in the numbers, we get 
$$k = \dfrac{|7 - 8| - 0.5(15)}{\sqrt{0.25 (15)}} = \dfrac{0.5}{0.967} \approx 0.517.$$
Since this is a two-tailed test, we need to compare the test statistic to the critical values at $= .025$ and $= .975$ with 14 degrees of freedom. From the table, we can see that the critical values are -2.145 and 2.145. Since 0.517 is smaller than 2.145, we fail to reject the null hypothesis at $= .05$. Therefore, we do not have enough evidence to conclude that the median distance is not 200 miles. The correct answer is A.

A Researcher Collects Data on the Distance Between College Students 252530304545557585110135175195810217025253030454555758511013517519581021702525303045455575851101351751958102170

A Researcher Collects Data on the Distance Between College Students $2525303045455575851101351751958102170$