Chapter PIV, Problem 28RE

(a)

To determine

To find: the mean difference in melon weights.

Answer to Problem 28RE

4 pounds

Explanation of Solution

Given:

Suppose

$X_1$ The weight of the melon at the first store represents

$E\left(X_1\right)=22$ pounds,$X_2$ The weight of the melon in the second store represents $E\left(X_2\right)=18$ pounds;

The melon 's standard deviation at the 1st store $SD\left(X_1\right)=2.5$ pounds;

The melon 's standard deviation at the 2nd store $SD\left(X_2\right)=2$ pounds

Formula used:

  $E\left(X_1-X_2\right)=E\left(X_1\right)-E\left(X_2\right)$

Calculation:

Mean difference in melon weights

  $\begin{array}{c}E\left(X_1-X_2\right)=E\left(X_1\right)-E\left(X_2\right)\\ =22-18\\ =4\text{Pounds}\end{array}$

(b)

To determine

To find: the standard deviation of the weights of the difference.

Answer to Problem 28RE

10.25

Explanation of Solution

Given:

Suppose

  $X_1$ The weight of the melon at the first store represents $E\left(X_1\right)=22$ pounds,$X_2$ The weight of the melon in the second store represents $E\left(X_2\right)=18$ pounds;

The melon 's standard deviation at the 1st store $SD\left(X_1\right)=2.5$ pounds;

The melon 's standard deviation at the 2nd store $SD\left(X_2\right)=2$ pounds

Formula used:

  $V\left(X_1-X_2\right)=V\left(X_1\right)+V\left(X_2\right)$

Calculation:

Standard Deviation of the difference in weights

  $\begin{array}{c}V\left(X_1-X_2\right)=V\left(X_1\right)+V\left(X_2\right)\\ =6.25+4\\ =10.25\end{array}$

(c)

To determine

To find: the probability of getting the melon at the 1^st store is heavier.

Answer to Problem 28RE

0.8925

Explanation of Solution

Given:

Suppose

$X_1$ The weight of the melon at the first store represents

$E\left(X_1\right)=22$ pounds,$X_2$ The weight of the melon in the second store represents $E\left(X_2\right)=18$ pounds;

The melon 's standard deviation at the 1st store $SD\left(X_1\right)=2.5$ pounds;

The melon 's standard deviation at the 2nd store $SD\left(X_2\right)=2$ pounds

Formula used:

  $Z=\frac{X-\mu }{\sigma }$

Calculation:

The probability of the melon got at the first store being heavier

  $\begin{array}{c}Z=\frac{X-\mu }{\sigma }\\ =\frac{0-4}{3.21}\\ =1.246\end{array}$

  $\begin{array}{c}P\left(z>-1.246\right)=1-P\left(z<-1.246\right)\\ =1-0.1075\\ =0.8925\end{array}$