Final answer:
The null hypothesis for the GPA and major study states that there is no difference in average GPAs across the three groups, and based on the ANOVA output, the conclusion is that there is no significant difference in GPAs. The survey was answered by 198 students.
Step-by-step explanation:
Hypothesis Testing for GPA Across Majors
(a) Writing the hypotheses for testing the difference in average GPA across majors involves establishing a null hypothesis (H0) and an alternative hypothesis (Ha). The null hypothesis typically states that there is no effect or no difference, whereas the alternative hypothesis states that there is an effect or a difference.
- H0: μ1 = μ2 = μ3 (The average GPAs of students in Arts and Humanities, Natural Sciences, and Social Sciences are equal)
- Ha: At least one of the means (μ1, μ2, μ3) is different (The average GPAs are not all equal across majors)
(b) Conclusion of the hypothesis test: Since the p-value (0.8313) is greater than the common alpha level of 0.05, we fail to reject the null hypothesis. This indicates that there is no statistically significant difference in average GPA across the three groups of majors.
(c) Sample size calculation: The sample size is the sum of the number of observations in all groups. Given that there are 2 degrees of freedom (Df) for the major groups, this suggests there are 3 groups (since Df for groups is one less than the number of groups). The Df for residuals is 195, which represents the number of observations minus the number of groups. Hence, the total sample size is 195 + 3 = 198 students who answered the survey question.