Step 1 :For this study, we should use z-test for a population proportion.
Step 2 :The null and alternative hypotheses would be: \(H_0: p = 0.51\) and \(H_1: p \neq 0.51\).
Step 3 :The test statistic is calculated as (sample proportion - population proportion) / standard error. The standard error is calculated as \(\sqrt{(population proportion * (1 - population proportion)) / sample size}\). The test statistic \(z = 1.645\).
Step 4 :The p-value is the probability of observing a test statistic as extreme as the one we calculated, assuming the null hypothesis is true. The p-value is \(0.1000\).
Step 5 :The p-value is larger than the significance level of 0.01.
Step 6 :Based on this, we should fail to reject the null hypothesis.
Step 7 :Thus, the final conclusion is that the data suggest the population proportion is not significantly larger than \(51 \%\) at \(\alpha=0.01\), so there is not sufficient evidence to conclude that the population proportion of students who played intramural sports who received a degree within six years is larger than 51\%.
Step 8 :Interpret the p-value in the context of the study: The p-value of 0.1000 means that if the true population proportion of students who played intramural sports and received a degree within six years is 51%, there is a 10% chance of observing a sample proportion as extreme as the one we observed (or more extreme) just by random chance.