Chapter 11.7, Problem 17PPSE

Solution

To find: The standard deviation for each year is to be determined and are to be used to describe how farm income varied from 2001 to 2002.

The standard deviations for the years 2001 and 2002 are 2678.9 and 2758.99 respectively.

Since the second data set has a larger standard deviation, the data set is much more spread out than the data in the first data set.

Given information:

The given data sets are:

  

The mean income for the years 2001 and 2002 are 6706.86 and 6738.14 (millions of dollars) respectively.

Calculation:

To determine the variance, the following table is used:

$x$	$\overline{x}$	$x-\overline{x}$	${\left(x-\overline{x}\right)}^2$
10653	6706.86	3946.14	15572021
7979	6706.86	1272.14	1618340.2
7537	6706.86	830.14	689132.42
4723	6706.86	-1983.86	3935700.5
9221	6706.86	2514.14	6320899.9
2938	6706.86	-3768.86	14204306
3897	6706.86	-2809.86	7895313.2
			50235713

Hence,

  $\begin{array}{l}{\sigma }^2=\frac{{\displaystyle \sum {\left(x-\overline{x}\right)}^2}}{n}\\ {\sigma }^2=\frac{50235713}{7}\\ {\sigma }^2=7176530.41\end{array}$

The standard deviation is given by:

  $\begin{array}{l}\sigma =\sqrt{{\sigma }^2}\\ \sigma =\sqrt{7176530.41}\\ \sigma =2678.9\end{array}$

The standard deviation for the first data set is 2678.9.

To determine the variance, the following table is used:

$x$	$\overline{x}$	$x-\overline{x}$	${\left(x-\overline{x}\right)}^2$
10834	6738.14	4095.86	16776069
7862	6738.14	1123.86	1263061.3
7478	6738.14	739.86	547392.82
4402	6738.14	-2336.14	5457550.1
9589	6738.14	2850.86	8127402.7
3223	6738.14	-3515.14	12356209
3779	6738.14	-2959.14	8756509.5
			53284195

Hence,

  $\begin{array}{l}{\sigma }^2=\frac{{\displaystyle \sum {\left(x-\overline{x}\right)}^2}}{n}\\ {\sigma }^2=\frac{53284195}{7}\\ {\sigma }^2=7612027.84\end{array}$

The standard deviation is given by:

  $\begin{array}{l}\sigma =\sqrt{{\sigma }^2}\\ \sigma =\sqrt{7612027.84}\\ \sigma =2758.99\end{array}$

The standard deviation for the first data set is 2758.99.

Conclusion:

The standard deviations for the years 2001 and 2002 are 2678.9 and 2758.99 respectively.

Since the second data set has a larger standard deviation, the data set is much more spread out than the data in the first data set.