Question

Recent revenue shortfalls in a midwestern state led to a reduction in the state budget for higher education. To offset the reduction, the largest state university proposed a $25\%$tuition increase. It was determined that such an increase was needed simply to compensate for the lost support from the state. Separate random samples of $50$freshmen, $50$sophomores, $50$juniors, and $50$seniors from the university were asked whether they were strongly opposed to the increase, given that it was the minimum increase necessary to maintain the university's budget at current levels. Here are the results:

Which null hypothesis would be appropriate for performing a chi-square test?

a. The closer students get to graduation, the less likely they are to be opposed to tuition increases.

b. The mean number of students who are strongly opposed is the same for each of the $4$years.

c. The distribution of student opinion about the proposed tuition increase is the same for each of the $4$years at this university.

d. Year in school and student opinion about the tuition increase are independent in the sample.

e. There is an association between year in school and opinion about the tuition increase at this university.

Short answer

The correct option is (c) i.e. there is a distribution of students based on their opinion about the proposed tuition increase is the same for the four years at the university.

Step-by-step solution

Given information

We need to find out the correct option for the given data.

Explanation for correct option (c)

We know that

• A chi-square goodness-of-fit test will be used if we are only interested in the distribution of one variable.
• A chi-square test for homogeneity is used when we are interested in the distribution of two variables and there are several independent samples.
• A chi-square test for independence is used when we are interested in the distribution of two variables and there is only one sample.

A year and Strongly opposed are the two variables we're interested in. We should also notice that we have four independent samples, hence the chi-square test for homogeneity should be used.

A chi-square test of homogeneity's null hypothesis asserts that there is no difference in the categorical variable's distribution for each of the populations/treatments.

$H_0$represents that the distribution of opinion is the same.

$H_{\alpha }$ represents that the distribution of opinion is not the same.

We then note that answer choice is the correct response (c).

Explanation for incorrect option (a)

As it is clear from the above step that the opinion of students are the same which means that there is no difference in the opinion of students closer to graduation than others.

Therefore, option (a) is incorrect.

Explanation for incorrect option (b) 

From given data

We can interpret that the no. of students opposing is not the same which implies that the mean of the students opposing will not be the same.

Therefore,

Option (b) is incorrect.

Explanation for incorrect option (d)

From Step two:

It is clear that the opinion about the tuition increase depends on the given sample.

Therefore,

Option (d) is incorrect option.

Explanation for incorrect option (e)

We know that

There is no association between a year in school and opinion about the tuition increase as in step three we clearly state that the opinion of every student group is the same.

Therefore,

Option (e) is incorrect option.