Questions: The monthly incomes for 12 randomly selected people, each with a bachelor's degree in economics, are shown on the right. Complete parts (a) through (c) below. Assume the population is normally distributed. 4450.58 4596.45 4366.49 4455.12 4151.39 3727.71 4283.72 4527.21 4407.63 3946.19 4023.82 4221.79 (a) Find the sample mean. The sample mean is 4263.2 (Round to one decimal place as needed.) (b) Find the sample standard deviation. The sample standard deviation is (Round to one decimal place as needed.)

The monthly incomes for 12 randomly selected people, each with a bachelor's degree in economics, are shown on the right. Complete parts (a) through (c) below. Assume the population is normally distributed. 4450.58 4596.45 4366.49 4455.12 4151.39 3727.71 4283.72 4527.21 4407.63 3946.19 4023.82 4221.79 (a) Find the sample mean. The sample mean is 4263.2 (Round to one decimal place as needed.) (b) Find the sample standard deviation. The sample standard deviation is (Round to one decimal place as needed.)
Transcript text: The monthly incomes for 12 randomly selected people, each with a bachelor's degree in economics, are shown on the right. Complete parts (a) through (c) below. Assume the population is normally distributed. \begin{tabular}{lll|} 4450.58 & 4596.45 & 4366.49 \\ 4455.12 & 4151.39 & 3727.71 \\ 4283.72 & 4527.21 & 4407.63 \\ 3946.19 & 4023.82 & 4221.79 \end{tabular} (a) Find the sample mean. $\bar{x}=4263.2$ (Round to one decimal place as needed.) (b) Find the sample standard deviation. $s=\square$ $\square$ (Round to one decimal place as needed.)
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Solution

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Solution Steps

Step 1: Calculate the Sample Mean

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

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

where \( N = 12 \) is the number of observations and \( \sum_{i=1}^N x_i = 51158.1 \). Thus,

\[ \bar{x} = \frac{51158.1}{12} = 4263.2 \]

Step 2: Calculate the Sample Variance

The sample variance \( s^2 \) is calculated using the formula:

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

Substituting the values, we find:

\[ s^2 = 67583.2 \]

Step 3: Calculate the Sample Standard Deviation

The sample standard deviation \( s \) is the square root of the variance:

\[ s = \sqrt{s^2} = \sqrt{67583.2} = 260.0 \]

Final Answer

The results are as follows:

  • Sample Mean: \( \bar{x} = 4263.2 \)
  • Sample Standard Deviation: \( s = 260.0 \)

Thus, the final answers are:

\[ \boxed{\bar{x} = 4263.2} \] \[ \boxed{s = 260.0} \]

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