Question

The tax auditor is selecting a sample of 6 tax return for an audit. if 3 or more of these returns are "improper," the entire population of 55 tax return will be audited. Complete parts(a)through (d)

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

(a) x has a hypergeometric distribution with N = 55, M = 15, n = 6

P(x: N, n, M) = C(M, x) * C(N - M, n - x) / C(N, n)

x	P(x) = C(15, x) * C(55 - 15, 6 - x) / C(55, 6)
0	0.1324
1	0.3405
2	0.3310
3	0.1551
4	0.0367
5	0.0041
6	0.0002

P(x ≥ 3) = P(3) + P(4) + P(5) + P(6) = 0.1961

(b) x has a hypergeometric distribution with N = 55, M = 20, n = 6

P(x: N, n, M) = C(M, x) * C(N - M, n - x) / C(N, n)

x	P(x) = C(20, x) * C(55 - 20, 6 - x) / C(55, 6)
0	0.0560
1	0.2240
2	0.3432
3	0.2574
4	0.0994
5	0.0187
6	0.0013

P(x ≥ 3) = P(3) + P(4) + P(5) + P(6) = 0.3769

(c) x has a hypergeometric distribution with N = 55, M = 5, n = 6

P(x: N, n, M) = C(M, x) * C(N - M, n - x) / C(N, n)

x	P(x) = C(5, x) * C(55 - 5, 6 - x) / C(55, 6)
0	0.5482
1	0.3654
2	0.0794
3	0.0068
4	0.0002
5	0.0000

P(x ≥ 3) = P(3) + P(4) + P(5) = 0.0070

(d) x has a hypergeometric distribution with N = 55, M = 10, n = 6

P(x: N, n, M) = C(M, x) * C(N - M, n - x) / C(N, n)

x	P(x) = C(10, x) * C(55 - 10, 6 - x) / C(55, 6)
0	0.2810
1	0.4214
2	0.2313
3	0.0587
4	0.0072
5	0.0004
6	0.0000

P(x ≥ 3) = P(3) + P(4) + P(5) + P(6)= 0.0663

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