Chapter PIV, Problem 30RE

To determine

To find: melons chosen randomly at every store can charge for the mean and standard deviation of the difference in price.

Explanation of Solution

Formula used:

  $\begin{array}{c}E\left(D\right)=E\left(32X-25Y\right)\\ \mathrm{var}\left(D\right)=\mathrm{var}\left(32X-25Y\right)\\ SD\left(D\right)=\sqrt{\mathrm{var}\left(D\right)}\end{array}$

Calculation:

Let X be the weight of the first store's watermelon.

Y is the weight of the second store's watermelon.

D is the difference in the amount that was paid.

  $D=32X-25Y$

The mean of difference is

  $\begin{array}{c}E\left(D\right)=E\left(32X-25Y\right)\\ =32E\left(X\right)-25E\left(Y\right)\\ =32\left(22\right)-25\left(18\right)\\ =704-450\\ =254\text{cents}\end{array}$

We first find the variable.

  $\begin{array}{c}\mathrm{var}\left(D\right)=\mathrm{var}\left(32X-25Y\right)\\ ={32}^2\mathrm{var}\left(X\right)+{25}^2\mathrm{var}\left(Y\right)\\ ={32}^2{\left(2.5\right)}^2+25{\left(2\right)}^2\\ =6400+2500\\ =8900\end{array}$

Standard deviation of difference is

  $\begin{array}{c}SD\left(D\right)=\sqrt{\mathrm{var}\left(D\right)}\\ =\sqrt{8900}\\ =94.34\end{array}$