Questions: The grade point averages (GPA) for 12 randomly selected college students are shown on the right. Complete parts (a) through (c) below. Assume the population is normally distributed. 2.4 3.5 2.8 1.7 0.9 4.0 2.4 1.1 3.7 0.4 2.4 3.1 (a) Find the sample mean. x̄=2.37 (Round to two decimal places as needed.) (b) Find the sample standard deviation. s= (Round to two decimal places as needed.)

Solution Steps

To find the sample mean \( \bar{x} \), we use the formula:

\[ \bar{x} = \frac{\sum_{i=1}^N x_i}{N} \]

where \( N \) is the number of observations and \( x_i \) are the individual data points. For our dataset:

\[ \sum_{i=1}^{12} x_i = 28.4 \quad \text{and} \quad N = 12 \]

Thus, the sample mean is calculated as:

\[ \bar{x} = \frac{28.4}{12} = 2.37 \]

The sample standard deviation \( s \) is calculated using the formula:

\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]

First, we find the variance \( s^2 \):

\[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = 1.32 \]

Then, we take the square root to find the standard deviation:

\[ s = \sqrt{1.32} \approx 1.15 \]

Final Answer