Question

New legislation passed in 2017 by the U.S. Congress changed tax laws that affect how many people file their taxes in 2018 and beyond. These tax law changes will likely lead many people to seek tax advice from their accountants (The New York Times). Backen and Hayes LLC is an accounting firm in New York state. The accounting firms believe that it may have to hire additional accountants to assist with the increased demand in tax advice for the upcoming tax season. Backen and Hayes LLC has developed the following probability distribution for x= number of new clients seeking tax advice.

x	f(x)
20	.05
25	.20
30	.25
35	.15
40	.15
45	.10
50	.10

a. Is this a valid probability distribution? Explain.

b. What is the probability that Backens and Hayes LLC will obtain 40 or more new clients?

c. What is the probability that Backens and Hayes LLC will obtain fewer than 35 new clients?

d. Compute the expected value, variance, and standard deviation of x.

Answer 1

a. Is this a valid probability distribution? Explain.

Answer: Yes, this is a valid probability distribution, because sum of all probabilities is one.

x	f(x)
20	0.05
25	0.2
30	0.25
35	0.15
40	0.15
45	0.1
50	0.1
Total	1

b. What is the probability that Backens and Hayes LLC will obtain 40 or more new clients?

Solution:

P(X≥40) = P(X=40) + P(X=45) + P(X=50)

P(X≥40) = 0.15 + 0.1 + 0.1

P(X≥40) = 0.35

Required probability = 0.35

c. What is the probability that Backens and Hayes LLC will obtain fewer than 35 new clients?

Solution:

P(X<35) = P(X=20) + P(X=25) + P(X=30)

P(X<35) = 0.05 + 0.20 + 0.25

P(X<35) = 0.50

d. Compute the expected value, variance, and standard deviation of x.

Mean = expected value = ∑XP(X)

Variance = ∑P(X)*(X – mean)^2

SD = sqrt(Variance)

The required calculation table is given as below:

X	P(X)	XP(X)	P(X)*(X - mean)^2
20	0.05	1	10.153125
25	0.2	5	17.1125
30	0.25	7.5	4.515625
35	0.15	5.25	0.084375
40	0.15	6	4.959375
45	0.1	4.5	11.55625
50	0.1	5	24.80625
Total	1	34.25	73.1875

Mean = expected value = ∑XP(X) = 34.25

Variance = ∑P(X)*(X – mean)^2

Variance = 73.1875

Standard deviation = sqrt(73.1875)

Standard deviation = 8.554969