Question

What are the chances that a person who is murdered actually knew the murderer? The answer to this question explains why a lot of police detective work begins with relatives and friends of the victim! About 70% of people who are murdered actually knew the person who committed the murder.† Suppose that a detective file in New Orleans has 60 current unsolved murders. Find the following probabilities. (Round your answers to four decimal places.)

(c) fewer than 30 victims did not know their murderers


(d) more than 20 victims did not know their murderers

Answer 1

P ( did not know the murderers ) = 0.3

P ( know the murderers ) = 0.7

Using Normal Approximation to Binomial
Mean = n * P = ( 60 * 0.3 ) = 18
Variance = n * P * Q = ( 60 * 0.3 * 0.7 ) = 12.6
Standard deviation = √(variance) = √(12.6) = 3.5496

Part a)
P ( X < 30 )
Using continuity correction
P ( X < n - 0.5 ) = P ( X < 30 - 0.5 ) = P ( X < 29.5 )

X ~ N ( µ = 18 , σ = 3.5496 )
P ( X < 29.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 29.5 - 18 ) / 3.5496
Z = 3.24
P ( ( X - µ ) / σ ) < ( 29.5 - 18 ) / 3.5496 )
P ( X < 29.5 ) = P ( Z < 3.24 )
P ( X < 29.5 ) = 0.9994

Part b)
P ( X > 20 )
Using continuity correction
P ( X > n + 0.5 ) = P ( X > 20 + 0.5 ) = P ( X > 20.5 )

X ~ N ( µ = 18 , σ = 3.5496 )
P ( X > 20.5 ) = 1 - P ( X < 20.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 20.5 - 18 ) / 3.5496
Z = 0.7
P ( ( X - µ ) / σ ) > ( 20.5 - 18 ) / 3.5496 )
P ( Z > 0.7 )
P ( X > 20.5 ) = 1 - P ( Z < 0.7 )
P ( X > 20.5 ) = 1 - 0.758
P ( X > 20.5 ) = 0.2420