Question

Dottie's Tax Service specializes in federal tax returns for professional clients, such as physicians, dentists, accountants, and lawyers. A recent audit by the IRS of the returns she prepared indicated that an error was made on 15% of the returns she prepared last year. Assuming this rate continues into this year and she prepares 52 returns, what is the probability that she makes errors on:

More than 8 returns? (Round your z-score computation to 2 decimal places and final answer to 4 decimal places.)

At least 8 returns? (Round your z-score computation to 2 decimal places and final answer to 4 decimal places.)

Exactly 8 returns? (Round your z-score computation to 2 decimal places and final answer to 4 decimal places.)

Answer 1

Solution:

What is the probability that she makes errors on More than 8 returns?

We are given n = 52, p = 0.15, q = 1 – p = 1 – 0.15 = 0.85

np = 52*0.15 = 7.8

nq = 52*0.85 = 44.2

np and nq > 5, so we can use normal approximation.

Mean = np = 52*0.15 = 7.8

SD = sqrt(npq) =sqrt(52*0.15*0.85) = 2.574879

Here, we have to find P(X>8)

P(X>8) ≈ P(X>8.5) (by using continuity correction)

P(X>8.5) = 1 – P(X<8.5)

Z = (X – mean) / SD

Z = (8.5 - 7.8) / 2.574879

Z = 0.27186

Z = 0.27

P(Z<0.27) = P(X<8.5) = 0.6064

(by using z-table)

P(X>8.5) = 1 – P(X<8.5)

P(X>8.5) = 1 – 0.6064

P(X>8.5) = 0.3936

Required probability = 0.3936

What is the probability that she makes errors on at least 8 returns?

Here, we have to find P(X≥8)

P(X≥8) = P(X>7.5) (by using continuity correction)

P(X>7.5) = 1 – P(X<7.5)

Z = (X – mean) / SD

Z = (7.5 - 7.8) / 2.574879

Z = -0.1165

Z = -0.12

P(Z<-0.12) = P(X<7.5) = 0.4522

(by using z-table)

P(X>7.5) = 1 – P(X<7.5)

P(X>7.5) = 1 – 0.4522

P(X>7.5) = 0.5478

Required probability = 0.5478

What is the probability that she makes errors on exactly 8 returns?

Here, we have to find P(X=8)

P(X=8) = P(7.5<X<8.5)

(by using continuity correction)

P(7.5<X<8.5) = P(X<8.5) – P(X<7.5)

From part a and b, we have

P(Z<0.27) = P(X<8.5) = 0.6064

P(Z<-0.12) = P(X<7.5) = 0.4522

P(7.5<X<8.5) = P(X<8.5) – P(X<7.5)

P(7.5<X<8.5) = 0.6064 - 0.4522

P(7.5<X<8.5) = 0.1542

Required probability = 0.1542