Chapter 2, Problem 2.40E

a.

To determine

To choose: Four numbers that have the smallest possible standard deviation.

Answer to Problem 2.40E

The four numbers chosen would have same values to get the smallest standard deviation.

Explanation of Solution

Given info:

From the whole numbers 0 to 10, four numbers are chosen with repetition.

Justification:

Standard deviation:

Formula for standard deviation is,

$s=\sqrt{\frac{1}{n-1}{\displaystyle \sum {\left(x_i-\overline{x}\right)}^2}}$

Standard deviation is always zero or greater than zero. If the observations have the same value with no variability then the standard deviation is zero. The standard deviation increases as the variability increases.

For example, the sets $\left(1,1,1,1\right),\left(2,2,2,2\right)$ have the same values and there is no variability between them.

Thus, the four number set with the same values have the smallest possible standard deviation.

b.

To determine

To choose: Four numbers that have the largest possible standard deviation.

Answer to Problem 2.40E

The four numbers that have the largest standard deviation is $\left(0,0,10,10\right)$.

Explanation of Solution

Calculation:

Suppose a set of four numbers $\left(0,0,10,10\right)$ are taken.

Mean:

The sample size n is 4.

Formula for mean is:

$\overline{x}=\frac{{\displaystyle \sum x_i}}{n}$

The mean value for the set $\left(0,0,10,10\right)$ is

$\begin{array}{c}\overline{x}=\frac{0+0+10+10}{4}\\ =\frac{20}{4}\\ =5\end{array}$

Thus, the mean is 5.

Standard deviation:

Formula for standard deviation is:

$s=\sqrt{\frac{1}{n-1}{\displaystyle \sum {\left(x_i-\overline{x}\right)}^2}}$

The standard deviation for the set $\left(0,0,10,10\right)$ is:

$\begin{array}{c}s=\sqrt{\frac{1}{4-1}\left[{\left(0-5\right)}^2+{\left(0-5\right)}^2+{\left(10-5\right)}^2+{\left(10-5\right)}^2\right]}\\ =\sqrt{\frac{1}{3}\left[{\left(-5\right)}^2+{\left(-5\right)}^2+{\left(5\right)}^2+{\left(5\right)}^2\right]}\\ =\sqrt{\frac{1}{3}\left[25+25+25+25\right]}\\ =\sqrt{\frac{100}{3}}\end{array}$

$\begin{array}{c}=\sqrt{33.3333}\\ =5.7735\end{array}$

Thus, the standard deviation is 5.77.

Suppose a set of four numbers $\left(0,10,10,10\right)$ are taken.

Mean:

The sample size n is 4.

Formula for mean is:

$\overline{x}=\frac{{\displaystyle \sum x_i}}{n}$

The mean value for the set $\left(0,10,10,10\right)$ is

$\begin{array}{c}\overline{x}=\frac{0+10+10+10}{4}\\ =\frac{30}{4}\\ =7.5\end{array}$

Thus, the mean is 7.5.

Standard deviation:

Formula for standard deviation is:

$s=\sqrt{\frac{1}{n-1}{\displaystyle \sum {\left(x_i-\overline{x}\right)}^2}}$

The standard deviation for the set $\left(0,10,10,10\right)$ is:

$\begin{array}{c}s=\sqrt{\frac{1}{4-1}\left[{\left(0-7.5\right)}^2+{\left(10-7.5\right)}^2+{\left(10-7.5\right)}^2+{\left(10-7.5\right)}^2\right]}\\ =\sqrt{\frac{1}{3}\left[{\left(-7.5\right)}^2+{\left(2.5\right)}^2+{\left(2.5\right)}^2+{\left(2.5\right)}^2\right]}\\ =\sqrt{\frac{1}{3}\left[56.25+6.25+6.25+6.25\right]}\\ =\sqrt{\frac{75}{3}}\end{array}$

$\begin{array}{c}=\sqrt{25}\\ =5\end{array}$

Thus, the standard deviation is 5.

From the whole numbers 0 to 10, the set $\left(0,10,10,10\right)$ have a small standard deviation. The set $\left(0,0,10,10\right)$ have the largest standard deviation and as the variability increases.

Thus, the four numbers that have the largest standard deviation is $\left(0,0,10,10\right)$.

c.

To determine

To check: Whether more than one choice is possible in either part (a) or part (b).

Answer to Problem 2.40E

More than one choice is possible in part (a) but not in part (b).

Explanation of Solution

From part (a), the least possible standard deviation is 0. Since, the whole numbers 0 to 10 can be repeatedly taken in a set of four numbers; there is more than one choice possible in part (a). But in part (b), the highest standard deviation is found and there is only one way to get using the whole numbers from 0 to 10 with repetition.

Thus, more than one choice is possible in part (a).