Question

Al Bundy is evaluating a new advertising program that could increase shoe sales. Possible outcomes and probabilities of the outcomes are shown next. Compute the coefficient of variation.

Possible Outcomes

Additional Sales

in Units Probabilities

Ineffective campaign ........................ 40 0.20

Normal response .............................. 60 0.50

Extremely effective ........................... 140 0.30

Short answer

Cofficient of variance is 0.5.

Step-by-step solution

Compute expected value-

$\begin{array}{c}\mathrm{Expected}\mathrm{value}=\mathrm{Respective}\mathrm{sales}\times \mathrm{Respective}\mathrm{probability}\\ \mathbf{=}(\mathbf{40}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{20})\mathbf{+}(\mathbf{60}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{50})\mathbf{+}(\mathbf{140}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{30})\\ =8+30+42\\ =80\mathrm{units}\end{array}$

Computation of total probability

Probability	Sales	Probability*(sales-Expected sales)2	Total
0.20	40	0.20*(40-80)2	320
0.50	60	0.50*(60-80)2	200
0.30	140	0.30*(140-80)2	1080
			1600

Computation of Standard deviation ( approx )-

$\begin{array}{l}\mathbf{Standard}\mathbf{}\mathbf{deviation}\mathbf{=}\sqrt{\begin{array}{l}\mathbf{Probability}\mathbf{\times }{(\mathbf{sales}\mathbf{-}\mathbf{Expectedsales})}^2\end{array}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{1600}\\ =40\mathrm{units}\end{array}$

Computation of co-efficient of variation ( approx.)-

$\begin{array}{c}\mathbf{Coefficient}\mathbf{}\mathbf{of}\mathbf{}\mathbf{variation}\mathbf{=}\frac{\mathrm{Standard}\mathrm{deviation}}{\mathrm{Expected}\mathrm{sales}}\\ \mathbf{=}\frac{40}{80}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5}\end{array}$