Question

sample annual salaries (in thousands of dollars) for employees at a company are listed.
49 42 46 54 40 40 49 42 46 32 54 49 45

(a) find the sample mean and sample standard deviation.
(b) each employee in the sample is given a \\$3000 raise. find the sample mean and sample standard deviation for the revised data set.
(c) each employee in the sample takes a pay cut of \\$5000 from their original salary. 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 \bar{x} = \box thousand dollars.
(round to one decimal place as needed.)

Explanation:

Calculate the sample mean

Using the Mean Calculation knowledge point

$$
LATEXBLOCK0
$$

Calculate the sample standard deviation

Using the Sample Standard Deviation knowledge point

$$
LATEXBLOCK1
$$

Apply linear transformations

Using the Linear Transformation of Data knowledge point

• For part (b), adding \(3\) to each value:

$$ \bar{x}_{\text{new}} = \bar{x} + 3 \approx 48.2,\quad s_{\text{new}} = s \approx 6.1 $$

• For part (c), subtracting \(5\) from each value:

$$ \bar{x}_{\text{new}} = \bar{x} - 5 \approx 40.2,\quad s_{\text{new}} = s \approx 6.1 $$

• For part (d), adding or subtracting a constant changes the mean by that constant but leaves the standard deviation unchanged.

Answer:

The sample mean is \(\bar{x} =\) <blank>45.2</blank> thousand dollars.