Question

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?

Answer 1

(a) We know that in a particular set of transactions, 2% contain an error of some kind. Hence if X is the random variable defined as: X= number of transactions found to be in error in a sample of 20,

Probability Distribution of X = Binomial with success probability p = 0.02 and number of trials n = 20.

P[X=2] = Combin(20,2) * (0.02)^2 * (1-0.02)^18

= 190*0.0004 * 0.695135 = 0.05283

Probability that 2 transactions are found to be in error = 0.05283

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?

P[X>=3] = 1- P[X<3] = 1- { P[X=0] + P[X=1] + P[X=2] }

Using the formula from (a),

P[X=0] = Combin(20,0) * (0.02)^0 * (1-0.02)^20

P[X=1] = Combin(20,1) * (0.02)^1 * (1-0.02)^19

P[X=2] = Combin(20,2) * (0.02)^2 * (1-0.02)^18

Evaluate these in Excel and we get

P[X=0] =

0.667607972

P[X=1] =

0.27249305

P[X=2] =  

0.05283

Therefore P[X=0] + P[X=1] + P[X=2] =

0.992931

Therefore P[X>=3] = 1- 0.992931 =

0.007069

Hence  three or more transactions are found to be in error happens only 0.707% of times

(c) What assumption is required for the validity of the above answers?

Assumptions: Sampling of 20 is done without replacement, but we assume that the total population of transactions N is sufficiently large. So that, we can use Binomial distribution for the sample of 20 and do not have to use the Hypergeometric distribution.