Internal auditors sometimes check random samples of transactions within a database. Suppose that in a particular set of transactions, 2% contain an error of some kind. The auditor takes a random sample of 20 transactions for checking. Let X denote the number of transactions found to be in error in the sample.
(a) State the probability distribution of X (including the values of all parameters) and find the probability that 2 transactions are found to be in error.
(b) If three or more transactions are found to be in error then a larger sample is taken for checking. How often will this happen? (Use the appropriate template).
(c) What assumption is required for the validity of the above answers?
Data
Sample size 100
Probability of success 0.2
Statistics
Mean 20
Variance 16
Standard deviation 4
Binomial Probabilities Table
X P(X) Cumulative
0 0.00000 0.00000
1 0.00000 0.00000
2 0.00000 0.00000
3 0.00000 0.00000
4 0.00000 0.00000
5 0.00001 0.00002
6 0.00006 0.00008
7 0.00020 0.00028
8 0.00058 0.00086
9 0.00148 0.00233
10 0.00336 0.00570
11 0.00688 0.01257
12 0.01275 0.02533
13 0.02158 0.04691
14 0.03353 0.08044
15 0.04806 0.12851
16 0.06383 0.19234
17 0.07885 0.27119
18 0.09090 0.36209
19 0.09807 0.46016
20 0.09930 0.55946
21 0.09457 0.65403
22 0.08490 0.73893
23 0.07198 0.81091
24 0.05773 0.86865
25 0.04388 0.91252
26 0.03164 0.94417
27 0.02168 0.96585
28 0.01413 0.97998
29 0.00877 0.98875
30 0.00519 0.99394
31 0.00293 0.99687
32 0.00158 0.99845
33 0.00081 0.99926
34 0.00040 0.99966
35 0.00019 0.99985
36 0.00009 0.99994
37 0.00004 0.99998
38 0.00002 0.99999
39 0.00001 1.00000
40 0.00000 1.00000
41 0.00000 1.00000
42 0.00000 1.00000
43 0.00000 1.00000
44 0.00000 1.00000
45 0.00000 1.00000
46 0.00000 1.00000
47 0.00000 1.00000
48 0.00000 1.00000
49 0.00000 1.00000
50 0.00000 1.00000
51 0.00000 1.00000
52 0.00000 1.00000
53 0.00000 1.00000
54 0.00000 1.00000
55 0.00000 1.00000
56 0.00000 1.00000
57 0.00000 1.00000
58 0.00000 1.00000
59 0.00000 1.00000
60 0.00000 1.00000
61 0.00000 1.00000
62 0.00000 1.00000
63 0.00000 1.00000
64 0.00000 1.00000
65 0.00000 1.00000
66 0.00000 1.00000
67 0.00000 1.00000
68 0.00000 1.00000
69 0.00000 1.00000
70 0.00000 1.00000
71 0.00000 1.00000
72 0.00000 1.00000
73 0.00000 1.00000
74 0.00000 1.00000
75 0.00000 1.00000
76 0.00000 1.00000
77 0.00000 1.00000
78 0.00000 1.00000
79 0.00000 1.00000
80 0.00000 1.00000
81 0.00000 1.00000
82 0.00000 1.00000
83 0.00000 1.00000
84 0.00000 1.00000
85 0.00000 1.00000
86 0.00000 1.00000
87 0.00000 1.00000
88 0.00000 1.00000
89 0.00000 1.00000
90 0.00000 1.00000
91 0.00000 1.00000
92 0.00000 1.00000
93 0.00000 1.00000
94 0.00000 1.00000
95 0.00000 1.00000
96 0.00000 1.00000
97 0.00000 1.00000
98 0.00000 1.00000
99 0.00000 1.00000
100 0.00000 1.00000