A university has six colleges and takes a poll to gauge student support for a tuition increase. The university wants to ensure each college is represented fairly. The below table shows the observed number of students that participate in the poll from each college and the actual proportion of students in each college. Using the critical value approach, the decision and conclusion are ________.
A) reject the null hypothesis; at least one of the proportions is different from its hypothesized value
B) reject the null hypothesis; all of the proportions are not the same
C) do not reject the null hypothesis; all proportions are equal to 0.20
D) do not reject the null hypothesis; we cannot conclude not all of the proportions are the same

Question

A university has six colleges and takes a poll to gauge student support for a tuition increase. The university wants to ensure each college is represented fairly. The below table shows the observed number of students that participate in the poll from each college and the actual proportion of students in each college.  Using the critical value approach, the decision and conclusion are ________.
A) reject the null hypothesis; at least one of the proportions is different from its hypothesized value 
B) reject the null hypothesis; all of the proportions are not the same 
C) do not reject the null hypothesis; all proportions are equal to 0.20 
D) do not reject the null hypothesis; we cannot conclude not all of the proportions are the same

Quizplus · Accepted Answer

To use the critical value approach, we need to calculate the expected number of students in each college if the proportion of students supporting a tuition increase is the same in each college. We can do this by multiplying the total number of students who participated in the poll (800) by the hypothesized proportion of 0.20 for each college. The expected number of students in each college is shown in the table below:

College  Observed Number  Expected Number
1        80              800 x 0.20 = 160
2        120             800 x 0.20 = 160
3        170             800 x 0.20 = 160
4        130             800 x 0.20 = 160
5        150             800 x 0.20 = 160
6        150             800 x 0.20 = 160

The degrees of freedom for the chi-square test of independence are calculated as (r-1) x (c-1), where r is the number of rows (in this case, 6) and c is the number of columns (in this case, 2). Therefore, the degrees of freedom are (6-1) x (2-1) = 5.

Using a chi-square calculator, we find the test statistic to be 28.9 and the p-value to be less than 0.001. Using a significance level of 0.05 and 5 degrees of freedom, the critical value for the chi-square distribution is 11.07.

Since the test statistic (28.9) is greater than the critical value (11.07) and the p-value is less than 0.001, we reject the null hypothesis. This means that there is evidence to suggest that at least one of the proportions is different from its hypothesized value.

A University Has Six Colleges and Takes a Poll to Gauge