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. (See "Who Loses HOPE? Attrition from Georgia's College Scholarship Program" (Southern Economic Journal [1999]: \(379-390\) ).) It was reported that \(53.2 \%\) of a random sample of 137 students entering Ivan Allen College at Georgia Tech (social science and human- ities) with a HOPE scholarship lost the scholarship at the end of the first year because they had a GPA of less than 3.0. It was also reported that 72 of a random sample of 111 students entering the College of Computing with a \(\mathrm{B}\) average had lost their HOPE scholarship by the end of the first year. Is there evidence that the proportion who lose HOPE scholarships is different for the College of Computing than for the Ivan Allen College?

Short answer

There is a significant difference in the proportion of students who lost the HOPE scholarship in the College of Computing compared to the Ivan Allen College

Step-by-step solution

Define the Null and Alternate hypothesis

The Null Hypothesis, H0: \(p_1 - p_2 = 0\) and Alternate Hypothesis, Ha: \(p_1 - p_2 ≠ 0\). Here, \(p_1\) denotes the population proportion for IAC and \(p_2\) denotes the population proportion for College of Computing.

Calculate the sample proportions

Calculate the sample proportions by using the formula \( \hat{p} = x/n \), For IAC: \( \hat{p_1} = 53.2\% = 0.532 \)For Computing: \( \hat{p_2} = 72/111 = 0.6486 \)

Calculate the pooled proportion

Pooled proportion (\(\hat{P}\)) is calculated as \( \hat{P} = (x1 + x2) / (n1 + n2) = \( (0.532*137 + 72) / (137 + 111) = 0.59 \)

Estimate the standard error

Standard error, SE is found using the formula \( SE = \sqrt{ \hat{P}(1 - \hat{P})(1/n1 + 1/n2) } = \sqrt{ 0.59(1 -0.59)(1/137 + 1/111) } = 0.055 \)

Calculate the z-value

The z-value is calculated using the formula \( z = (\hat{p1} - \hat{p2}) / SE = (0.532 - 0.6486) / 0.055 = -2.118 \)

Obtain the p-value and state conclusion

Using the standard normal distribution table or calculator, the two-tailed P-value was found to be about 0.034. As this P-value is less than 0.05, reject the null hypothesis. Thus there is enough evidence to suggest that the proportion of students who lost the HOPE scholarship is different for the College of Computing than for Ivan Allen College.