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 $41.0 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.)

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, we need to find the mean $\bar{x}$. Assume the data - points are $x_1,x_2,\cdots,x_n$. The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.

Step2: Calculate the mean

Since the problem doesn't provide the actual data values in the table, let's assume the data values are $x_1,x_2,\cdots,x_8$ (because $n = 8$). Calculate $\bar{x}=\frac{x_1 + x_2+\cdots+x_8}{8}$.

Step3: Calculate $(x_{i}-\bar{x})^{2}$ for each $i$

For $i = 1,2,\cdots,8$, calculate $(x_{i}-\bar{x})^{2}$. Then find $\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}$.

Step4: Calculate the sample standard deviation

Using the formula $s=\sqrt{\frac{\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}}{8 - 1}}=\sqrt{\frac{\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}}{7}}$.

Since the data values are not given in the problem description, we can't perform the actual calculations. But if we had the data values, we would follow the above steps.

If we assume the data values are $x_1,x_2,\cdots,x_8$ and we calculate:
Let $\sum_{i = 1}^{8}x_{i}=S$, then $\bar{x}=\frac{S}{8}$.
Let $d_i=x_i-\bar{x}$ for $i = 1,\cdots,8$. Then $\sum_{i = 1}^{8}d_{i}^{2}=D$.
The sample standard deviation $s=\sqrt{\frac{D}{7}}$.

Answer:

(Insufficient data to provide a numerical answer. If the data values in the table were provided, we could calculate the sample standard deviation following the above steps and round to one decimal place as needed.)