Question

Gender differences in student needs and fears were examined in the article "A Survey of Counseling Needs of Male and Female College Students" (Journal of College Student Development \([1998]: 205-208\) ). Random samples of male and female students were selected from those attending a particular university. Of 234 males surveyed, \(27.5 \%\) said that they were concerned about the possibility of getting AIDS. Of 568 female students surveyed, \(42.7 \%\) reported being concerned about the possibility of getting AIDS. Is there sufficient evidence to conclude that the proportion of female students concerned about the possibility of getting AIDS is greater than the corresponding proportion for males?

Short answer

The conclusion whether there is sufficient evidence to conclude that the proportion of female students concerned about the possibility of getting AIDS is greater than the corresponding proportion for males can be made based on the p-value and the significance level. If the p-value is less than the significance level, then there is sufficient evidence to reject the null hypothesis and conclude that the proportion of female students concerned about the possibility of getting AIDS is greater than that of males.

Step-by-step solution

Set Up Hypotheses

The null hypothesis \(H_0\) is that the proportions are equal, i.e. \(P_F = P_M\). The alternative hypothesis \(H_A\) is that the proportion of concern among female students is greater than among male students, i.e. \(P_F > P_M\).

Calculate Test Statistic

Calculate the test statistic using the formula for comparing two population proportions: \(Z = \frac{(\hat{P}_F - \hat{P}_M) - 0}{\sqrt{\hat{P}(1-\hat{P})\left(\frac{1}{n_F} + \frac{1}{n_M}\right)}}\) where \(\hat{P}_F = 42.7\%, \hat{P}_M = 27.5\%, n_F = 568, n_M = 234\), and \(\hat{P}\) is the overall sample proportion, \(\hat{P} = \frac{n_F\hat{P}_F + n_M\hat{P}_M}{n_F + n_M}\). Substitute these values to calculate the Z score.

Find P-Value

Find the p-value associated with the test statistic. The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. Use a standard normal table or calculator to find the p-value.

Make Decision

Compare the p-value with the significance level. If the p-value is less than the significance level, reject the null hypothesis. If the p-value is greater than or equal to the significance level, fail to reject the null hypothesis.