Question

• Put each problem on a separate worksheet in an Excel file. Type the tabular data into the spreadsheet
• Make Excel do all of the necessary calculations
• Report answers on the worksheet
• Label answers as the questions are labelled: a, b, c,…
• Use the probability notation p(A), p(Aand B),… etc

1. Each salesperson in a large department store chain is rated on their sales ability and their potential for advancement. The data for the 500 sampled salespeople are summarized in the following table.

	Potential for Advancement
Sales Ability	Fair	Good	Excellent
Below Average	16	12	22
Average	45	60	45
Above Average	93	72	135

a. What is the probability that a randomly selected salesperson has above average sales ability?

b. What is the probability that a randomly selected salesperson has a good chance for advancement?

c. What is the probability that a randomly selected salesperson has average sales ability and a good chance for advancement?

d. What is the probability that a randomly selected salesperson has average sales ability or a good chance for advancement?

e. Ferris White has had great results over the past 6 months and is considered to have Above average sales ability. What is the probability that his potential for advancement is Excellent?

f. It Is known that the owner’s son has Excellent potential for advancement. What is the probability that he has below average sales ability?

g.   Are the events Sales Ability and Potential for Advancement independent? Show the test.

Answer 1

Solution:

Given:The data for the 500 sampled salespeople are summarized in the following table.

	Potential for Advancement
Sales Ability	Fair	Good	Excellent	Total
Below Average	16	12	22	50
Average	45	60	45	150
Above Average	93	72	135	300
Total	154	144	202	N = 500

Part a. What is the probability that a randomly selected salesperson has above average sales ability?

P( above average sales ability ) =..............?

Let A = above average sales ability

thus

P(A) = Total of above average sales ability / N

P(A) = 300 / 500

P(A) = 0.6

Part b. What is the probability that a randomly selected salesperson has a good chance for advancement?

Let B = salesperson has a good chance for advancement

Thus

P(B) = Total of good chance for advancement / N

P(B) = 144 / 500

P(B) = 0.288

Part c. What is the probability that a randomly selected salesperson has average sales ability and a good chance for advancement?

P(salesperson has average sales ability and a good chance for advancement ) =..............?

Let C = average sales ability and B = salesperson has a good chance for advancement

Thus we have to find:

P( C and B ) = .........?

P( C and B ) = Number of salesperson has average sales ability and a good chance for advancement / N

P( C and B ) = 60 / 500

P( C and B ) = 0.12

Thus P(salesperson has average sales ability and a good chance for advancement ) = 0.12

Part d. What is the probability that a randomly selected salesperson has average sales ability or a good chance for advancement?

P( Salesperson has average sales ability or a good chance for advancement) = .........?

P( C or B) =.............?

Using addition rule of probability:

P(C or B) = P(C) + P(B) - P( C and B)

We have P(B) = 0.288 , P( C and B ) = 0.12

Now find

P(C) = P( average sales ability)

P(C) = Total of average sales ability / N

P(C) = 150 / 500

P(C) = 0.3

Thus we get:

P(C or B) = P(C) + P(B) - P( C and B)

P(C or B) = 0.3 + 0.288 - 0.12

P(C or B) = 0.468

P( Salesperson has average sales ability or a good chance for advancement) = 0.468

Part e. Ferris White has had great results over the past 6 months and is considered to have Above average sales ability. What is the probability that his potential for advancement is Excellent?

Let D =  potential for advancement is Excellent and A = Above average sales ability

Thus we have to find:

P( potential for advancement is Excellent | have Above average sales ability) =..........?

P( D | A) =...........?

where

P( D and A) = Number of salesperson has above average sales ability and potential for advancement is Excellent / N

P( D and A) = 135 / 500

P( D and A) = 0.27

and we have:

P(A) = 0.6

Thus we get:

Part f. It Is known that the owner’s son has Excellent potential for advancement. What is the probability that he has below average sales ability?

Let E = Salesperson has below average sales ability

D =  potential for advancement is Excellent

Find:

P( E | D) = ...........?

where

P( E and D) = Number of salesperson has below average sales ability and potential for advancement is Excellent / N

P( E and D) = 22 / 500

P( E and D) = 0.044

and

P(D) = P( potential for advancement is Excellent)

P(D) = Total of potential for advancement is Excellent / N

P(D) = 202 / 500

P(D) = 0.404

Thus

Part g. Are the events Sales Ability and Potential for Advancement independent? Show the test.

the events Sales Ability and Potential for Advancement are independent if following condition satisfied for each:

P( Sales Ability and Potential for Advancement ) = P( Sales Ability ) X P( Potential for Advancement )

Lets consider:

Sales Ability = average sales ability and Potential for Advancement = good chance for advancement

thus we have:

P( salesperson has average sales ability and a good chance for advancement ) = P( C and B) = 0.12

P(C) = P( average sales ability) = 0.3

P(B) =P(good chance for advancement) = 0.288

Thus find:

P(C) X P(B) = 0.3 X 0.288

P(C) X P(B) = 0.0864 which is not equal to P( C and B) = 0.12

Thus the events Sales Ability and Potential for Advancement are not independent.