Question

annual salaries (in thousands of dollars) for employees at a company are listed. 58 54 36 36 35 52 58 31 54 35 40... the sample mean and sample standard deviation... each employee in the sample is given a 5% raise. find the sample mean and sample standard deviation for the... st data set. calculate the monthly salary, divide each original salary by 12. find the sample mean and sample standard deviation for the revised data set. what can you conclude from the results of (a), (b), and (c)? what can you conclude from the results of (a), (b), and (c)? a. when each entry is multiplied by a constant k, the new sample mean is k•x̄ and the sample standard deviation remains unaffected. b. when each entry is multiplied by a constant k, the sample mean and the sample standard deviation remain unaffected. c. when each entry is multiplied by a constant k, the new sample standard deviation is k•s and the sample mean remains unaffected. d. when each entry is multiplied by a constant k, the new sample mean is k•x̄ and the new sample standard deviation is k•s.

Explanation:

To determine the correct conclusion, we analyze the effect of multiplying each data entry by a constant \( k \) on the sample mean (\( \bar{x} \)) and sample standard deviation (\( s \)).

• The sample mean is calculated as \( \bar{x} = \frac{\sum x_i}{n} \). If each \( x_i \) is multiplied by \( k \), the new sum is \( \sum (k x_i) = k \sum x_i \), so the new mean \( \bar{x}' = \frac{k \sum x_i}{n} = k \bar{x} \).
• The sample standard deviation is calculated as \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} \). If each \( x_i \) is multiplied by \( k \), the new deviations are \( (k x_i - k \bar{x}) = k(x_i - \bar{x}) \), so the new squared deviations are \( k^2(x_i - \bar{x})^2 \). The new standard deviation \( s' = \sqrt{\frac{\sum k^2(x_i - \bar{x})^2}{n - 1}} = k \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} = k s \).

This matches option D: "When each entry is multiplied by a constant \( k \), the new sample mean is \( k \cdot \bar{x} \) and the new sample standard deviation is \( k \cdot s \)."

Answer:

D. When each entry is multiplied by a constant \( k \), the new sample mean is \( k \cdot \bar{x} \) and the new sample standard deviation is \( k \cdot s \).