Step 1 :Define the null and alternative hypothesis: \(H_{0}: \mu=3\) and \(H_{1}: \mu \neq 3\)
Step 2 :This is a two-tailed test because we are testing for a difference and not a specific direction (greater or less than).
Step 3 :Given that the sample mean (\(\bar{x}\)) is 2.98, the standard deviation (s) is 0.08, and the sample size (n) is 20.
Step 4 :Calculate the test statistic (t) using the formula: \(t = (\bar{x} - \mu) / (s / \sqrt{n})\)
Step 5 :Substitute the given values into the formula to get \(t = (2.98 - 3) / (0.08 / \sqrt{20})\), which gives \(t = -1.12\)
Step 6 :Find the p-value corresponding to the calculated t-value. The p-value is approximately 0.28.
Step 7 :Since the p-value (0.28) is greater than the significance level (0.2), we fail to reject the null hypothesis.
Step 8 :This means that we do not have enough evidence to support the claim that the mean GPA of night students is significantly different than 3.
Step 9 :Final Answer: We \(\boxed{\text{fail to reject the null hypothesis}}\). The mean GPA of night students is not significantly different than 3 at the 0.2 significance level.