Question

each year, forbes magazine compiles a list of the 400 richest americans. as of september 19, 2012, 8 of the top 10 on the list are as shown in the accompanying table. determine the range and sample standard deviation. click the icon to view the table of 8 of the top 10 richest americans. the range is $41 billion. (type an integer or a decimal. do not round.) the sample standard deviation is $ billion. (type an integer or a decimal. round to one decimal place as needed.) top 10 richest people in a country person wealth ($ billions) bill gates 66.0 warren buffett 46.0 david koch 31.0 charles koch 31.0 christy walton and family 27.9 alice walton 26.3 s. robson walton 26.1 michael bloomberg 25.0

Explanation:

Step1: Recall sample - standard deviation formula

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$, where $x_{i}$ are the data - points, $\bar{x}$ is the sample mean, and $n$ is the sample size.
First, find the sample mean $\bar{x}$.
$n = 8$
$\bar{x}=\frac{66.0 + 46.0+31.0 + 31.0+27.9+26.3+26.1+25.0}{8}$
$=\frac{279.3}{8}=34.9125$

Step2: Calculate $(x_{i}-\bar{x})^{2}$ for each $x_{i}$

For $x_1 = 66.0$: $(66.0 - 34.9125)^{2}=(31.0875)^{2}=966.43$
For $x_2 = 46.0$: $(46.0 - 34.9125)^{2}=(11.0875)^{2}=122.93$
For $x_3 = 31.0$: $(31.0 - 34.9125)^{2}=(- 3.9125)^{2}=15.31$
For $x_4 = 31.0$: $(31.0 - 34.9125)^{2}=(-3.9125)^{2}=15.31$
For $x_5 = 27.9$: $(27.9 - 34.9125)^{2}=(-7.0125)^{2}=49.18$
For $x_6 = 26.3$: $(26.3 - 34.9125)^{2}=(-8.6125)^{2}=74.17$
For $x_7 = 26.1$: $(26.1 - 34.9125)^{2}=(-8.8125)^{2}=77.67$
For $x_8 = 25.0$: $(25.0 - 34.9125)^{2}=(-9.9125)^{2}=98.25$

Step3: Calculate $\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}$

$\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=966.43+122.93 + 15.31+15.31+49.18+74.17+77.67+98.25$
$=1419.25$

Step4: Calculate the sample standard deviation

$s=\sqrt{\frac{1419.25}{8 - 1}}=\sqrt{\frac{1419.25}{7}}\approx\sqrt{202.75}\approx14.2$

Answer:

14.2