Sample annual salaries​ (in thousands of​ dollars) for employees at a company are listed.
54  39  50  62  40  40  54  39  50  26  62  54  43   
​(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 ​$5000 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)?


### (a) Find the sample mean and sample standard deviation.

First, we need to calculate the **sample mean** and **sample standard deviation** of the given salaries. The sample data (in thousands of dollars) is:

\[ 54, 39, 50, 62, 40, 40, 54, 39, 50, 26, 62, 54, 43 \]

- **Sample Mean**:
  \[
  \bar{x} = \frac{\sum{x_i}}{n}
  \]
  where $x_i$ is each salary value, and $n$ is the number of values (13 in this case).

- **Sample Standard Deviation**:
  \[
  s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n-1}}
  \]

Let's compute both.

### (b) After a $4000 raise for each employee

If each employee receives a raise of **$4000**, the data points are adjusted by adding **4** (since the values are in thousands). This means the new data set is:
\[
58, 43, 54, 66, 44, 44, 58, 43, 54, 30, 66, 58, 47
\]

Now, compute the **sample mean** and **sample standard deviation** of this new data set.

### (c) After a $5000 pay cut for each employee

If each employee takes a pay cut of **$5000**, the data points are adjusted by subtracting **5** (since the values are in thousands). The new data set is:
\[
49, 34, 45, 57, 35, 35, 49, 34, 45, 21, 57, 49, 38
\]

We then calculate the **sample mean** and **sample standard deviation** for this adjusted data.

### (d) Conclusions from (a), (b), and (c)

We will observe the effects of adding and subtracting a constant amount (either a raise or a cut) on the mean and standard deviation.

Let me compute the mean and standard deviation for each case now.Here are the results for each part:

### (a) Original dataHere are the results for each part:

### (a) Original 
- **Sample Mean**: $ 47.15 $ thousand dollars
- **Sample Standard Deviation**: $ 10.40 $ thousand dollars

### (b) After a $4000 raise for each employee:
- **Sample Mean**: $ 51.15 $ thousand dollars
- **Sample Standard Deviation**: $ 10.40 $ thousand dollars

### (c) After a $5000 pay cut for each employee:
- **Sample Mean**: $ 42.15 $ thousand dollars
- **Sample Standard Deviation**: $ 10.40 $ thousand dollars

### (d) Conclusions:
- The **mean** changes when a constant amount is added or subtracted to each salary, but the change is exactly equal to the added (or subtracted) amount. Specifically, a $4000 raise increases the mean by 4, and a $5000 cut decreases the mean by 5.
- The **standard deviation** remains unchanged. This is because adding or subtracting the same amount to all data points shifts the entire data set but does not affect the spread of the data.

Would you like more details or explanations on any of these steps?

Here are some related questions to explore further:
1. How does multiplying each salary by a constant affect the mean and standard deviation?
2. What would the median salary be in each scenario?
3. How would removing the highest and lowest salaries affect the mean and standard deviation?
4. What is the variance of the original salary data?
5. How can we interpret the standard deviation in terms of salary distribution?

**Tip**: Adding or subtracting a constant shifts the data's central tendency (mean) but doesn't affect its dispersion (standard deviation).

Sample annual salaries​ (in thousands of​ dollars) for employees at a company are listed.
54  39  50  62  40  40  54  39  50  26  62  54  43.
​(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 ​$5000 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)?

Learn how to calculate the sample mean and standard deviation of a company's salary data. This example demonstrates the effects of raises and pay cuts on salary distributions. Suitable for high school statistics students.

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