It cannot be statistically distinguished between the current sample's average GPA (3.53) and the hypothesized population average (3.27). This suggests that the observed difference might be due to chance, and we cannot confidently claim that the current applicant pool has a significantly different average GPA than the historical average.
What can be concluded after testing the hypothesis?
Hypothesis: The current sample of 37 applicants to the graduate program comes from a population with an average GPA of 3.27.
Null Hypothesis: H 0: μ = 3.27
Alternative Hypothesis: H a: μ ≠ 3.27 (non-directional)
Significance Level: α = 0.05
Test Statistic: Since we don't know the population standard deviation (σ), we'll use a one-sample t-test.
t = (x- μ) / (s / √n)
t = (3.53 - 3.27) / (0.29 / √37)
t ≈ 1.03
Using a t-distribution table with 36 degrees of freedom (n-1), we find the two-tailed p-value for t = 1.03 to be approximately 0.318.
Since the p-value (0.318) is greater than the significance level (0.05), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the current sample comes from a population with a different average GPA than 3.27.
Based on the data, we cannot statistically distinguish between the current sample's average GPA (3.53) and the hypothesized population average (3.27). This suggests that the observed difference might be due to chance, and we cannot confidently claim that the current applicant pool has a significantly different average GPA than the historical average.