Questions: 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? The range is million. (Round to the nearest integer as needed.)

Solution Steps

The range of a dataset is the difference between the maximum and minimum values. For the given net worths:

\[ \text{Range} = \max(5500, 3200, 1700, 1000, 1000, 950) - \min(5500, 3200, 1700, 1000, 1000, 950) = 5500 - 950 = 4550 \]

The mean \(\mu\) is calculated as:

\[ \mu = \frac{\sum x_i}{n} = \frac{5500 + 3200 + 1700 + 1000 + 1000 + 950}{6} = \frac{13350}{6} = 2225.0 \]

The variance \(\sigma^2\) for a sample is calculated using:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \]

Calculating each squared deviation:

\[ (5500 - 2225)^2 = 1071225, \quad (3200 - 2225)^2 = 950625, \quad (1700 - 2225)^2 = 275625 \] \[ (1000 - 2225)^2 = 151225, \quad (1000 - 2225)^2 = 151225, \quad (950 - 2225)^2 = 1625625 \]

Sum of squared deviations:

\[ 1071225 + 950625 + 275625 + 151225 + 151225 + 1625625 = 16578750 \]

Variance:

\[ \sigma^2 = \frac{16578750}{5} = 3315750.0 \]

The standard deviation is the square root of the variance:

\[ \sigma = \sqrt{3315750.0} = 1820.92 \]

Final Answer