Chapter 16, Problem 43E

(a)

To determine

To Explain: Random variables to convey the farmer's net profits by using them.

Explanation of Solution

Given:

Suppose that

X is the cost of pound of apple.

Y is the cost of pound of potatoes.

	Mean	SD
X	0.5	0.2
Y	0.3	0.1

Calculation:

Suppose N farmers total income

  $N=100X+50Y-2$

(b)

To determine

To find: the mean.

Answer to Problem 43E

$63

Explanation of Solution

Given:

Suppose that

X is the cost of pound of apple.

Y is the cost of pound of potatoes.

	Mean	SD
X	0.5	0.2
Y	0.3	0.1

Calculation:

Compute the Mean net income

  $\begin{array}{c}E\left(N\right)=E\left(100X+50Y-2\right)\\ =100E\left(X\right)+50E\left(Y\right)-2\\ =100\left(0.5\right)+50\left(0.3\right)-2\\ =50+15-2\\ =\$63\end{array}$

(c)

To determine

To find: the standard deviation of the entire income.

Answer to Problem 43E

$20.62

Explanation of Solution

Given:

Suppose that

X is the cost of pound of apple.

Y is the cost of pound of potatoes.

	Mean	SD
X	0.5	0.2
Y	0.3	0.1

Formula used:

  $\begin{array}{c}\mathrm{var}\left(N\right)=\mathrm{var}\left(100X+50Y-2\right)\\ SD\left(N\right)=\sqrt{\mathrm{var}\left(N\right)}\end{array}$

Calculation:

Compute the standard deviation of the income.

First find the variance

  $\begin{array}{c}\mathrm{var}\left(N\right)=\mathrm{var}\left(100X+50Y-2\right)\\ ={100}^2\mathrm{var}\left(X\right)+{50}^2\mathrm{var}\left(Y\right)\\ ={100}^2\left(0.2\right)+{50}^2\left(0.1\right)\\ =425\end{array}$

Standard deviation:

  $\begin{array}{c}SD\left(N\right)=\sqrt{\mathrm{var}\left(N\right)}\\ =\sqrt{425}\\ =\$20.62\end{array}$

(d)

To determine

To Explain: that there is need to create any assumptions in estimating the mean and about the standard deviation.

Explanation of Solution

There is need to suppose that the cost of apples is independent of cost of potatoes in estimating the standard deviation