
Skip to Main content
Question 1
Question 2
Question 3
Question 4
Question 5
Question 6
Question 7
Question 8
Question 9
Question 10
Question 11
Question 12
Question content area top
Part 1
Here are 6 celebrities with some of the highest net worths​ (in millions of​ dollars) in a recent​ year: George Lucas ​(5500​), Oprah Winfrey ​(3200​), Michael Jordan ​(1700​), J. K. Rowling ​(1000​), David Copperfield ​(1000​), and Jerry Seinfeld ​(950​)   . Find the​ range, variance, and standard deviation for the sample data. What do the results tell us about the population of all​ celebrities? Based on the nature of the​ amounts, what can be inferred about their​ precision?

### Step 1: Organize the data
The net worths (in millions of dollars) are:

- George Lucas: \( 5500 \)
- Oprah Winfrey: \( 3200 \)
- Michael Jordan: \( 1700 \)
- J.K. Rowling: \( 1000 \)
- David Copperfield: \( 1000 \)
- Jerry Seinfeld: \( 950 \)

So, the data set is: \( 5500, 3200, 1700, 1000, 1000, 950 \).

### Step 2: Calculate the Range
The range is the difference between the largest and smallest values.

\[
\text{Range} = \text{max value} - \text{min value} = 5500 - 950 = 4550
\]

### Step 3: Calculate the Mean
The mean \( \bar{x} \) is the sum of all values divided by the number of data points.

\[
\bar{x} = \frac{5500 + 3200 + 1700 + 1000 + 1000 + 950}{6} = \frac{13350}{6} = 2225
\]

### Step 4: Calculate the Variance
The variance \( s^2 \) for a sample is calculated using:

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

Where:
- \( x_i \) are the data points,
- \( \bar{x} \) is the mean,
- \( n \) is the number of data points.

First, calculate \( (x_i - \bar{x})^2 \) for each value:

\[
(5500 - 2225)^2 = 1071225
\]
\[
(3200 - 2225)^2 = 950625
\]
\[
(1700 - 2225)^2 = 275625
\]
\[
(1000 - 2225)^2 = 1515625
\]
\[
(1000 - 2225)^2 = 1515625
\]
\[
(950 - 2225)^2 = 1625625
\]

Now, sum these squared differences:

\[
1071225 + 950625 + 275625 + 1515625 + 1515625 + 1625625 = 6956250
\]

Finally, divide by \( n - 1 = 6 - 1 = 5 \):

\[
s^2 = \frac{6956250}{5} = 1391250
\]

### Step 5: Calculate the Standard Deviation
The standard deviation \( s \) is the square root of the variance:

\[
s = \sqrt{1391250} \approx 1180.01
\]

### Results Summary
- **Range**: \( 4550 \) million dollars
- **Variance**: \( 1391250 \) million dollars squared
- **Standard Deviation**: \( 1180.01 \) million dollars

### Interpretation:
1. **Range**: The wide range of \( 4550 \) million dollars shows a significant difference in wealth among the top celebrities.
2. **Standard Deviation**: A high standard deviation of \( 1180.01 \) million dollars indicates that the data points (celebrity net worths) are spread out from the mean, meaning there is significant variability in wealth.
3. **Variance**: The large variance supports the conclusion that the wealth distribution among these individuals is very uneven.

### Precision:
Given the massive scale of the net worth values, we can infer that small differences in reported amounts (e.g., rounding) are not highly significant. The precision is less critical because the large differences (in the millions) dominate the analysis.

---

Would you like more details or have any questions?

### Related Questions:
1. How does the variance in this sample compare to a smaller or larger group of celebrities?
2. Can the median be a better measure of central tendency for such a data set?
3. How would the results change if one of the values was significantly higher or lower?
4. What effect would removing an outlier like George Lucas have on the standard deviation?
5. How can this information be used in business or media to interpret celebrity wealth trends?

### Tip:
When analyzing highly skewed data, like extreme wealth, consider using the median and interquartile range for a clearer picture of the distribution.

Here are 6 celebrities with some of the highest net worths​ (in millions of​ dollars) in a recent​ year: George Lucas ​(5500​), Oprah Winfrey ​(3200​), Michael Jordan ​(1700​), J. K. Rowling ​(1000​), David Copperfield ​(1000​), and Jerry Seinfeld ​(950​)   . Find the​ range, variance, and standard deviation for the sample data. What do the results tell us about the population of all​ celebrities? Based on the nature of the​ amounts, what can be inferred about their​ precision?

Explore how to calculate the range, variance, and standard deviation of the net worths of top celebrities including George Lucas, Oprah Winfrey, and Michael Jordan. This problem covers core statistical concepts such as sample variance and standard deviation, and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation