Freshmen GPAs First-semester GPAs for a random selection of freshmen at a large university are shown. Estimate the true mean GPA of the freshman class with $95 \%$ confidence. Assume $\sigma=0.62$. Round intermediate and final answers to two decimal places. Assume the population is normally distributed.
\begin{tabular}{lllllllll}
3.8 & 3.0 & 2.8 & 3.3 & 2.7 & 2.9 & 2.0 & 3.2 & 1.9 \\
2.8 & 2.2 & 4.0 & 1.9 & 2.8 & 2.0 & 2.7 & 3.9 & 3.8 \\
3.5 & 2.7 & 3.1 & 3.8 & 3.0 & 3.2 & 2.8 & 2.7 & 2.5 \\
1.9 & & & & & & & &
\end{tabular}

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\square< \mu< \square
\]
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Step 1 ：Given the GPAs of a random selection of freshmen at a large university, we are asked to estimate the true mean GPA of the freshman class with 95% confidence. The given GPAs are: 3.8, 3.0, 2.8, 3.3, 2.7, 2.9, 2.0, 3.2, 1.9, 2.8, 2.2, 4.0, 1.9, 2.8, 2.0, 2.7, 3.9, 3.8, 3.5, 2.7, 3.1, 3.8, 3.0, 3.2, 2.8, 2.7, 2.5, 1.9. We also know that the population standard deviation (\(\sigma\)) is 0.62.

Step 2 ：First, we calculate the sample mean of the given GPAs. The sample mean is the sum of all the values divided by the number of values. The sample mean (\(\overline{x}\)) is approximately 2.89.

Step 3 ：Next, we calculate the sample size (n), which is the number of GPAs given. In this case, n = 28.

Step 4 ：We then calculate the Z-score for the given confidence level (0.95). The Z-score is a measure of how many standard deviations an element is from the mean. For a confidence level of 0.95, the Z-score is approximately 1.96.

Step 5 ：We calculate the margin of error next. The margin of error is the range in which the true population mean is likely to be. It is calculated by multiplying the Z-score by the standard deviation divided by the square root of the sample size. The margin of error is approximately 0.23.

Step 6 ：Finally, we calculate the confidence interval, which is the range in which the true population mean is likely to be, with a certain level of confidence. The confidence interval is calculated by subtracting the margin of error from the sample mean and adding the margin of error to the sample mean. The confidence interval is approximately (2.66, 3.12).

Step 7 ：So, the estimated true mean GPA of the freshman class with 95% confidence is between 2.66 and 3.12. Therefore, the answer is \(\boxed{2.66<\mu<3.12}\)