Questions: 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 99% confidence. Assume σ=0.62. Round intermediate and final answers to two decimal places. Assume the population is normally distributed. 2.7, 3.1, 3.8, 3.0, 3.2, 2.8, 2.7, 2.5, 1.9, 2.0, 2.7, 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

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 99% confidence. Assume σ=0.62. Round intermediate and final answers to two decimal places. Assume the population is normally distributed.

2.7, 3.1, 3.8, 3.0, 3.2, 2.8, 2.7, 2.5, 1.9, 2.0, 2.7, 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

Transcript text: 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 $99 \%$ confidence. Assume $\sigma=0.62$. Round intermediate and final answers to two decimal places. Assume the population is normally distributed. \begin{tabular}{lllllllll} 2.7 & 3.1 & 3.8 & 3.0 & 3.2 & 2.8 & 2.7 & 2.5 & 1.9 \\ 2.0 & 2.7 & 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 \end{tabular}

Solution

Solution Steps

Step 1: Calculate the Mean GPA

The mean GPA for the freshmen is calculated as follows:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{78.3}{28} = 2.8 \]

Thus, the Mean GPA is $ 2.8 $.

Step 2: Determine the Z-Score

For a $ 99\% $ confidence level, the Z-score is:

\[ Z = 2.58 \]

Step 3: Calculate the Margin of Error

The margin of error is calculated using the formula:

\[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} = \frac{2.58 \times 0.62}{\sqrt{28}} \approx 0.3 \]

Thus, the Margin of Error is $ 0.3 $.

Step 4: Construct the Confidence Interval

The $ 99\% $ confidence interval is given by:

\[ \text{Confidence Interval} = \left( \mu - \text{Margin of Error}, \mu + \text{Margin of Error} \right) = \left( 2.8 - 0.3, 2.8 + 0.3 \right) = (2.5, 3.1) \]