Questions: Sample annual salaries (in thousands of dollars) for employees at a company are listed. 46, 47, 48, 52, 33, 33, 46, 47, 48, 27, 52, 46, 41 (a) Find the sample mean and sample standard deviation. (b) Each employee in the sample is given a 4000 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 4000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set. (d) What can you conclude from the results of (a), (b), and (c)? (Round to one decimal place as needed.) The sample standard deviation is s=7.8 thousand dollars. (Round to one decimal place as needed.) (b) The sample mean is x̄=47.5 thousand dollars. (Round to one decimal place as needed.) The sample standard deviation is s=7.8 thousand dollars. (Round to one decimal place as needed.) (c) The sample mean x̄=39.5 thousand dollars. (Round to one decimal place as needed.) The sample standard deviation is s=7.8 t thousand dollars. (Round to one decimal place as needed.) (d) What can you conclude from the results of (a), (b), and (c)? A. When a constant k is added to or subtracted from each entry, the new sample mean is x̄+k or x̄−k, respectively, and the sample standard deviation remains unaffected. B. When a constant k is added to or subtracted from each entry, the new sample mean is x̄+k or x̄−k, respectively, and the new sample standard deviation is s+k. C. When a constant k is added to or subtracted from each entry, the sample mean is unaffected, and the new sample standard deviation is s+k or s−k, respectively D. When a constant k is added to or subtracted from each entry, the new sample mean is x̄+k or x̄−k, respectively, and the new sample standard deviation is s · k

Sample annual salaries (in thousands of dollars) for employees at a company are listed. 46, 47, 48, 52, 33, 33, 46, 47, 48, 27, 52, 46, 41 (a) Find the sample mean and sample standard deviation. (b) Each employee in the sample is given a 4000 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 4000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set. (d) What can you conclude from the results of (a), (b), and (c)? (Round to one decimal place as needed.) The sample standard deviation is s=7.8 thousand dollars. (Round to one decimal place as needed.) (b) The sample mean is x̄=47.5 thousand dollars. (Round to one decimal place as needed.) The sample standard deviation is s=7.8 thousand dollars. (Round to one decimal place as needed.) (c) The sample mean x̄=39.5 thousand dollars. (Round to one decimal place as needed.) The sample standard deviation is s=7.8 t thousand dollars. (Round to one decimal place as needed.) (d) What can you conclude from the results of (a), (b), and (c)? A. When a constant k is added to or subtracted from each entry, the new sample mean is x̄+k or x̄−k, respectively, and the sample standard deviation remains unaffected. B. When a constant k is added to or subtracted from each entry, the new sample mean is x̄+k or x̄−k, respectively, and the new sample standard deviation is s+k. C. When a constant k is added to or subtracted from each entry, the sample mean is unaffected, and the new sample standard deviation is s+k or s−k, respectively D. When a constant k is added to or subtracted from each entry, the new sample mean is x̄+k or x̄−k, respectively, and the new sample standard deviation is s · k
Transcript text: Sample annual salaries (in thousands of dollars) for employees at a company are listed. $\begin{array}{lllllllllllll}46 & 47 & 48 & 52 & 33 & 33 & 46 & 47 & 48 & 27 & 52 & 46 & 41\end{array}$ (a) Find the sample mean and sample standard deviation. (b) Each employee in the sample is given a $\$ 4000$ 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 $\$ 4000$ from their original salary. Find the sample mean and the sample standard deviation for the revised data set. (d) What can you conclude from the results of (a), (b), and (c)? (Round to one decimal place as needed.) The sample standard deviation is $s=7.8$ thousand dollars. (Round to one decimal place as needed.) (b) The sample mean is $\bar{x}=47.5$ thousand dollars. (Round to one decimal place as needed.) The sample standard deviation is $s=7.8$ thousand dollars. (Round to one decimal place as needed.) (c) The sample mean $\bar{x}=39.5$ thousand dollars. (Round to one decimal place as needed.) The sample standard deviation is $s=7.8 \mathrm{t}$ thousand dollars. (Round to one decimal place as needed.) (d) What can you conclude from the results of (a), (b), and (c)? A. When a constant $k$ is added to or subtracted from each entry, the new sample mean is $\bar{x}+k$ or $\bar{x}-k$, respectively, and the sample standard deviation remains unaffected. B. When a constant $k$ is added to or subtracted from each entry, the new sample mean is $\bar{x}+k$ or $\bar{x}-k$, respectively, and the new sample standard deviation is $s+k$. C. When a constant $k$ is added to or subtracted from each entry, the sample mean is unaffected, and the new sample standard deviation is $s+k$ or $s-k$, respectively D. When a constant $k$ is added to or subtracted from each entry, the new sample mean is $\bar{x}+k$ or $\bar{x}-k$, respectively, and the new sample standard deviation is $s \cdot k$
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Solution

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Solution Steps

To solve the given problem, we will follow these steps:

(a) Calculate the sample mean and sample standard deviation for the given data set. (b) Adjust the data set by adding $4000$ to each salary, then calculate the new sample mean and sample standard deviation. (c) Adjust the data set by subtracting $4000$ from each salary, then calculate the new sample mean and sample standard deviation. (d) Analyze the results from (a), (b), and (c) to draw a conclusion about the effect of adding or subtracting a constant on the sample mean and standard deviation.

Step 1: Calculate the Sample Mean and Standard Deviation for the Original Data

Given the sample salaries: \[ \{46, 47, 48, 52, 33, 33, 46, 47, 48, 27, 52, 46, 41\} \]

The sample mean (\(\bar{x}\)) is calculated as: \[ \bar{x} = \frac{\sum x_i}{n} = \frac{566}{13} \approx 43.5385 \]

The sample standard deviation (\(s\)) is calculated as: \[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \approx 7.7848 \]

Step 2: Calculate the Sample Mean and Standard Deviation After a $4000 Raise

Each salary is increased by $4000, resulting in the new data set: \[ \{50, 51, 52, 56, 37, 37, 50, 51, 52, 31, 56, 50, 45\} \]

The new sample mean (\(\bar{x}_{\text{raise}}\)) is: \[ \bar{x}_{\text{raise}} = \bar{x} + 4 \approx 47.5385 \]

The new sample standard deviation (\(s_{\text{raise}}\)) remains the same: \[ s_{\text{raise}} = s \approx 7.7848 \]

Step 3: Calculate the Sample Mean and Standard Deviation After a $4000 Pay Cut

Each salary is decreased by $4000, resulting in the new data set: \[ \{42, 43, 44, 48, 29, 29, 42, 43, 44, 23, 48, 42, 37\} \]

The new sample mean (\(\bar{x}_{\text{cut}}\)) is: \[ \bar{x}_{\text{cut}} = \bar{x} - 4 \approx 39.5385 \]

The new sample standard deviation (\(s_{\text{cut}}\)) remains the same: \[ s_{\text{cut}} = s \approx 7.7848 \]

Final Answer

\(\boxed{\text{A}}\)

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