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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:

The conditions for carrying out the chi-square test in Exercise T$11.1$ are:

I. Independent random samples from the populations of interest.

II. All sample sizes are less than $10\%$of the populations of interest.

III. All expected counts are at least $5$.

Which of the conditions is (are) satisfied in this case?

a. I only

b. II only

c. I and III only

d. II and III only

e. I, II, and III

Step by step solution

01

Given information

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

02

Explanation

We know that

Random, independent ($10\%$), and large counts are the parameters for performing a chi-square test of homogeneity/independence.

I. Random: I'm satisfied because the samples are all random.

II. Self-sufficient: Content since the $50$pupils from each year represent less than $10\%$of all students (assuming that the university has at least $500$students in each year).

III. High counts: I'm happy because all of the projected counts are at least $5$.

As a result, all of the conditions are met, and (e) I, II, and III are right.

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