Question

The annual commissions earned by sales representatives of Machine Products Inc. a manufacturer of light machinery, follow the normal probability distribution. The mean yearly amount earned is $42,000 and the standard deviation is $4000.

a. What percent of representatives earn less than $45,000?

b. What percent of representatives earn between $32,000 and $45,000?

c. How much is the annual commission for the top 15% of the sales representatives?

Answer 1

Solution :

x : The annual commissions earned by sales representatives

µ = $42,000
σ = $4000

Using Normal Distribution,

a. What percent of representatives earn less than $45,000?

P(x<$45,000) = P((x-µ)/σ < (45,000-µ)/σ)

=P(z < (45,000-42000)/4000)

= P(z<0.75)

= 0.7734

Answer : 0.7734

b. What percent of representatives earn between $32,000 and $45,000?

P($32,000<x<$45,000) = P((x-µ)/σ < (45,000-µ)/σ) - P((x-µ)/σ < (32,000-µ)/σ)

=P(z < (45,000-42000)/4000) - P(z < (32,000-42000)/4000)

= P(z<0.75) - P(z<-2.50)

= 0.7734 - 0.0062

= 0.7672

Answer : 0.7672

c. How much is the annual commission for the top 15% of the sales representatives?

For top 15%,

Z = 1.04

the annual commission for the top 15% of the sales representatives = zσ + µ = 1.04*4000+42000 = 46160

Answer : 46160