A local company has received complaints that its telephone lines are always busy

Question

A local company has received complaints that its telephone lines are always busy. After examining thousands of company calls, an investigagtion into the complaints reveals that 60% of the telephone calls to the compnay are of a personal nature. Before the company can crack down on the personal call problem, the employees unionlodgtes a fomal complaint witht eh compnay. the union claims that the 60% number is outrageous and gives a sample of 20 random calls as evidence. Assuming the 60% number is accurate how many of the 20 calls shoudl we expect to be of a personal nature? Find the probability bthat at leasgt 16 of the calls were of a personal nature.

Explanation / Answer

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( X < 16) = P(X=15) + P(X=14) + P(X=13) + P(X=12) + P(X=11) ....P(X=1) + P(X=0)
= ( 20 15 ) * 0.6^15 * ( 1- 0.6 ) ^5 + ( 20 14 ) * 0.6^14 * ( 1- 0.6 ) ^6 + ( 20 13 ) * 0.6^13 * ( 1- 0.6 ) ^7 + ( 20 12 ) * 0.6^12 * ( 1- 0.6 ) ^8 + ( 20 11 ) * 0.6^11 * ( 1- 0.6 ) ^9 + ........... + ( 20 1 ) * 0.6^1 * ( 1- 0.6 ) ^19 + ( 20 0 ) * 0.6^0 * ( 1- 0.6 ) ^20
= 0.949

P(at leasgt 16 of the calls were of a personal nature) = P( X > = 16 ) = 1 - P( X < 16) = 0.051

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