Question

here are 6 celebrities with some of the highest net worths (in millions of dollars) in a recent year. george lucas (5500), steven spielberg (3700), oprah winfrey (3200), paul mccartney (1200), j. k. rowling (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 $4550 million. (round to the nearest integer as needed.) the variance is □ million dollars squared (round to the nearest integer as needed.)

Explanation:

Step1: Recall range formula

The range of a data - set is the difference between the maximum and minimum values. Given data values: 3700, 3200, 1200, 1000, 5500, 950. The maximum value is 5500 and the minimum value is 950.
Range = 5500 - 950 = 4550

Step2: Recall sample variance formula

The formula for the sample variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$, where $n$ is the sample size, $x_{i}$ are the data points, and $\bar{x}$ is the sample mean.
First, find the sample mean $\bar{x}=\frac{3700 + 3200+1200 + 1000+5500+950}{6}=\frac{15550}{6}\approx2591.67$
Then, calculate $(x_{1}-\bar{x})^{2},(x_{2}-\bar{x})^{2},\cdots,(x_{6}-\bar{x})^{2}$:
$(3700 - 2591.67)^{2}=(1108.33)^{2}=1228490.89$
$(3200 - 2591.67)^{2}=(608.33)^{2}=370064.89$
$(1200 - 2591.67)^{2}=(- 1391.67)^{2}=1936788.89$
$(1000 - 2591.67)^{2}=(-1591.67)^{2}=2533418.89$
$(5500 - 2591.67)^{2}=(2908.33)^{2}=8458360.89$
$(950 - 2591.67)^{2}=(-1641.67)^{2}=2695048.89$
The sum $\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=1228490.89+370064.89+1936788.89+2533418.89+8458360.89+2695048.89 = 17222173.34$
$s^{2}=\frac{17222173.34}{6 - 1}=\frac{17222173.34}{5}=3444434.67\approx3444435$

Answer:

The variance is 3444435 million dollars squared.