sample annual salaries (in thousands of dollars) for employees at a company are listed. 45 37 51 52 36 36 45…

sample annual salaries (in thousands of dollars) for employees at a company are listed. 45 37 51 52 36 36 45 37 51 30 52 45 50 (a) find the sample mean and sample standard deviation. (b) each employee in the sample is given a 6% raise. find the sample mean and sample standard deviation for the revised data set. to calculate the monthly salary, divide each original salary by 12. find the sample mean and sample standard deviation for the revised data set. (d) what can you conclude from the results of (a), (b), and (c)? (a) the sample mean is x = 43.6 thousand dollars. (round to one decimal place as needed.) (b) the sample mean is x = 46.2 thousand dollars. (round to one decimal place as needed.) the sample standard deviation is s = 7.6 thousand dollars. (round to one decimal place as needed.) (c) the sample mean is x = 3.6 thousand dollars. (round to one decimal place as needed.) the sample standard deviation is s = 42.33 thousand dollars. (round to one decimal place as needed.)

sample annual salaries (in thousands of dollars) for employees at a company are listed. 45 37 51 52 36 36 45 37 51 30 52 45 50 (a) find the sample mean and sample standard deviation. (b) each employee in the sample is given a 6% raise. find the sample mean and sample standard deviation for the revised data set. to calculate the monthly salary, divide each original salary by 12. find the sample mean and sample standard deviation for the revised data set. (d) what can you conclude from the results of (a), (b), and (c)? (a) the sample mean is x = 43.6 thousand dollars. (round to one decimal place as needed.) (b) the sample mean is x = 46.2 thousand dollars. (round to one decimal place as needed.) the sample standard deviation is s = 7.6 thousand dollars. (round to one decimal place as needed.) (c) the sample mean is x = 3.6 thousand dollars. (round to one decimal place as needed.) the sample standard deviation is s = 42.33 thousand dollars. (round to one decimal place as needed.)

Answer

Explanation:

Step1: Recall properties of mean and standard - deviation under transformation

If we have a data - set (x_1,x_2,\cdots,x_n) with mean (\bar{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i) and standard deviation (s=\sqrt{\frac{1}{n - 1}\sum_{i=1}^{n}(x_i-\bar{x})^2}), and we create a new data - set (y_i = ax_i + b) (in part (b), (a=\frac{1}{12}) and (b = 0); in part (c), (y_i=(1 + 0.06)x_i)).

Step2: Effect on the mean when dividing by a constant

If (y_i=\frac{1}{12}x_i), the new mean (\bar{y}=\frac{1}{12}\bar{x}). Given (\bar{x}=43.6) (from part (a)), then (\bar{y}=\frac{43.6}{12}\approx3.6).

Step3: Effect on the standard deviation when dividing by a constant

The standard deviation (s_y=\frac{1}{12}s_x). Given (s_x = 7.6) (from part (a)), then (s_y=\frac{7.6}{12}\approx0.6).

Step4: Effect on the mean when multiplying by a constant

If (y_i = 1.06x_i), the new mean (\bar{y}=1.06\bar{x}). Substituting (\bar{x}=43.6), we get (\bar{y}=1.06\times43.6 = 46.2).

Step5: Effect on the standard deviation when multiplying by a constant

The new standard deviation (s_y = 1.06s_x). Substituting (s_x = 7.6), we get (s_y=1.06\times7.6=8.056\approx8.1).

Step6: Conclusion for part (d)

When we divide each data - point by a constant (a), both the mean and the standard deviation are divided by (a). When we multiply each data - point by a constant (a), both the mean and the standard deviation are multiplied by (a).

Answer:

(a) The sample mean (\bar{x}=43.6) thousand dollars. (b) The sample mean is (\bar{x}=3.6) thousand dollars, and the sample standard deviation is (s = 0.6) thousand dollars. (c) The sample mean is (\bar{x}=46.2) thousand dollars, and the sample standard deviation is (s = 8.1) thousand dollars. (d) When we divide each value in a data - set by a non - zero constant (c), the sample mean and sample standard deviation are divided by (c). When we multiply each value in a data - set by a non - zero constant (k), the sample mean and sample standard deviation are multiplied by (k).