Questions: (c) Construct a sampling distribution for the mean by listing the sample means and their corresponding probabilities. Give all the unique sample means in ascending order. (Type integers or decimals. Type N if there is no solution.) Sample Mean Probability Sample Mean Probability 1 37.0 0.1 6 45.5 0.2 2 39.5 0.1 7 46.5 0.1 3 40.5 0.1 8 49 0.1 4 42 0.1 9 51.5 0.1 5 43 0.1 10 N N (d) Compute the mean of the sampling distribution. The mean of the sampling distribution is years.

Solution Steps

The mean of the sampling distribution is calculated using the formula:

\[ \text{Mean} = \sum (x_i \times p_i) \]

where \(x_i\) are the sample means and \(p_i\) are their corresponding probabilities. Substituting the values:

\[ \text{Mean} = 37.0 \times 0.1 + 39.5 \times 0.1 + 40.5 \times 0.1 + 42 \times 0.1 + 43 \times 0.1 + 45.5 \times 0.2 + 46.5 \times 0.1 + 49 \times 0.1 + 51.5 \times 0.1 \]

Calculating this gives:

\[ \text{Mean} = 44.0 \]

The variance is calculated using the formula:

\[ \text{Variance} = \sigma^2 = \sum ((x_i - \text{Mean})^2 \times p_i) \]

Substituting the values:

\[ \text{Variance} = (37.0 - 44.0)^2 \times 0.1 + (39.5 - 44.0)^2 \times 0.1 + (40.5 - 44.0)^2 \times 0.1 + (42 - 44.0)^2 \times 0.1 + (43 - 44.0)^2 \times 0.1 + (45.5 - 44.0)^2 \times 0.2 + (46.5 - 44.0)^2 \times 0.1 + (49 - 44.0)^2 \times 0.1 + (51.5 - 44.0)^2 \times 0.1 \]

Calculating this gives:

\[ \text{Variance} = 17.85 \]

The standard deviation is the square root of the variance:

\[ \text{Standard Deviation} = \sigma = \sqrt{\text{Variance}} = \sqrt{17.85} \approx 4.225 \]

Final Answer