Question

A tax auditor is selecting a sample of 5 tax returns for an audit. If 2 or more of these returns are "improper," the entire population of 55 tax returns will be audited. Q. What is the probability that the entire population will be audited if the true number of improper returns in the population is: a) 15 b) 20 c) 5 d) 10

Answer 1

Answer:

Given,

N = 55 , n = 5 , x >= 2

a)

K = 15

P(X >= 2) = 1 P(X < 2)

= 1 - [P(X=0) + P(X=1)]

= 1 - [15C0*((55-15)C(5-0)) / 55C5 + 15C1*(55-15)C(5-1) / 55C5]

= 1 - [1*40C5/55C5 + 15*40C4/55C5]

= 1 - [0.1892 + 0.0263]

= 1 - 0.2155

P(X >= 2) = 0.7845

b)

K = 20

P(X >= 2) = 1 P(X < 2)

= 1 - [P(X=0) + P(X=1)]

= 1 - [20C0*((55-20)C(5-0)) / 55C5 + 20C1*(55-20)C(5-1) / 55C5]

= 1 - [35C5/55C5 + 20*35C4/55C5]

= 1 - [0.0933 + 0.3010]

= 1 - 0.3943

= 0.6057

c)

K = 5

P(X >= 2) = 1 P(X < 2)

= 1 - [P(X=0) + P(X=1)]

= 1 - [5C0*((55-5)C(5-0)) / 55C5 + 5C1*(55-5)C(5-1) / 55C5]

= 1 - [50C5/55C5 + 5*50C4/55C5]

= 1 - [0.6091 + 0.3310]

= 1 - 0.9401

= 0.0599

d)

K = 10

P(X >= 2) = 1 P(X < 2)

= 1 - [P(X=0) + P(X=1)]

= 1 - [10C0*((55-10)C(5-0)) / 55C5 + 10C1*(55-10)C(5-1) / 55C5]

= 1 - [45C5/55C5 + 10*45C4/55C5]

= 1 - [0.3512 + 0.4283]

= 1 - 0.7795

= 0.2205