Question

The state of Georgia's HOPE scholarship program guarantees fully paid tuition to Georgia public universities for Georgia high school seniors who have a B average in academic requirements as long as they maintain a B average in college. Of 137 randomly selected students enrolling in the Ivan Allen College at the Georgia Institute of Technology (social science and humanities majors) in 1996 who had a B average going into college, \(53.2 \%\) had a GPA below \(3.0\) at the end of their first year ("Who Loses HOPE? Attrition from Georgia's College Scholarship Program," Southern Economic Journal [1999]: 379-390). Do these data provide convincing evidence that a majority of students at Ivan Allen College who enroll with a HOPE scholarship lose their scholarship?

Short answer

The conclusion depends on the comparison of the test statistic and the critical value. If the test statistic is greater than the critical value, there is evidence that a majority of students at Ivan Allen College who enroll with a HOPE scholarship lose their scholarship. If the test statistic is less than the critical value, there is not enough evidence to allege that a majority lose their scholarship.

Step-by-step solution

State the hypotheses

The first step in hypothesis testing is to specify the null hypothesis and the alternative hypothesis. The null hypothesis (Ho): The proportion of students at Ivan Allen College who lose their HOPE scholarship is \(50 \%\) or 0.5. This means that a majority of the students are not losing their scholarships.The alternative hypothesis (Ha): The proportion of students at Ivan Allen College who lose their HOPE scholarship is more than \(50 \%\) or 0.5. This implies that a majority of the students are losing their scholarships.

Calculate the Test Statistic

The test statistic in this case is a z-score. The formula for the z test statistic for a population proportion is \[ z = \frac{(p - p_{0})} {\sqrt{ \frac{(p_{0} \times (1 - p_{0}))} {n} }} \] where \( p \) is the sample proportion, \( p_{0} \) is the population proportion under the null hypothesis, and \( n \) is the sample size. In this case, \( p = 0.532 \), \( p_{0} = 0.5 \), and \( n = 137 \). Substitute these values into the formula and calculate the test statistic value.

Determine the critical value

The critical value corresponding to a level of significance \( \alpha = 0.05 \) for a one-tailed test and degrees of freedom \( df = n - 1 = 136 \) is approximately 1.645. This means that if the test statistic calculated is greater than 1.645, the null hypothesis will be rejected.

Compare the Test Statistic to the Critical Value

If the calculated test statistic is greater than the critical value, it is in the rejection region, so the null hypothesis can be rejected. If it's smaller, the null hypothesis cannot be rejected.