Sample annual salaries​ (in thousands of​ dollars) for employees at a company are listed.
50  43  61  48  35  35  50  43  61  28  48  50  42   
​(a) Find the sample mean and sample standard deviation.
​(b) Each employee in the sample is given a 66​% raise. Find the sample mean and sample standard deviation for the revised data set.
​(c) 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)?


Let's go through each part step-by-step:

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

#### Step 1: Calculate the Sample Mean
The sample mean \(\bar{x}\) is calculated by summing all the data points and dividing by the number of observations.

\[
\bar{x} = \frac{\sum x_i}{n}
\]

Where \( n \) is the number of observations, and \( x_i \) represents each salary value.

#### Step 2: Calculate the Sample Standard Deviation
The sample standard deviation \( s \) is calculated using the formula:

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

Let's compute these values.

### (b) Each employee in the sample is given a 66% raise. Find the sample mean and sample standard deviation for the revised data set.

When all values in a data set are multiplied by a constant \( k \), the new mean and standard deviation are given by:

\[
\text{New mean} = k \times \bar{x}
\]

\[
\text{New standard deviation} = |k| \times s
\]

In this case, \( k = 1.66 \).

### (c) To calculate the monthly salary, divide each original salary by 12. Find the sample mean and sample standard deviation for the revised data set.

When all values in a data set are divided by a constant \( k \), the new mean and standard deviation are given by:

\[
\text{New mean} = \frac{\bar{x}}{k}
\]

\[
\text{New standard deviation} = \frac{s}{|k|}
\]

Here, \( k = 12 \).

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

In general, multiplying or dividing the data set by a constant scales the mean and standard deviation accordingly. This helps to understand how transformations of data affect the measures of central tendency and variability.

Let's compute these values now.### Results

**(a) Original Sample:**
- **Mean:** 45.69 (thousands of dollars)
- **Standard Deviation:** 9.60 (thousands of dollars)

**(b) After 66% Raise:**
- **Mean:** 75.85 (thousands of dollars)
- **Standard Deviation:** 15.93 (thousands of dollars)

**(c) Monthly Salaries:**
- **Mean:** 3.81 (thousands of dollars)
- **Standard Deviation:** 0.80 (thousands of dollars)

### (d) Conclusions
1. **Multiplying by a constant (b)**: The mean and standard deviation are scaled by the same factor (1.66). This results in an increase in both the central tendency and variability of the data.
2. **Dividing by a constant (c)**: Both the mean and standard deviation are reduced by the same factor (12 in this case), reflecting a proportional reduction in the scale of the data.
3. **General Rule**: Scaling (multiplying or dividing) the entire dataset by a constant changes both the mean and standard deviation by that constant, but it does not affect the relative dispersion of the data.

Do you want more details or have any questions? Here are some related questions you might find interesting:

1. How does adding a constant to each salary affect the mean and standard deviation?
2. What happens to the variance when the data set is scaled by a constant?
3. Can the standard deviation ever be negative?
4. How does the median change with a 66% raise?
5. What would be the mode of the original and revised datasets?

**Tip:** Remember, the sample standard deviation uses \( n - 1 \) as the denominator, which corrects for bias in small samples.

Sample annual salaries​ (in thousands of​ dollars) for employees at a company are listed. 50  43  61  48  35  35  50  43  61  28  48  50  42 ​(a) Find the sample mean and sample standard deviation. ​(b) Each employee in the sample is given a 66​% raise. Find the sample mean and sample standard deviation for the revised data set. ​(c) 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)?

Learn how to compute the sample mean and standard deviation of a salary dataset, and explore how a 66% raise and monthly conversion affect these statistics. This problem covers key statistical concepts of mean and standard deviation with detailed transformations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation