Question

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

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 66% of people who are murdered actually knew the person who committed the murder.† Suppose that a detective file in New Orleans has 62 current unsolved murders. Find the following probabilities. (Round your answers to four decimal places.)

(a) at least 35 of the victims knew their murderers


(b) at most 48 of the victims knew their murderers


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


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

Answer 1

X ~ Bin ( n ,p)

Where n = 62 , p = 0.66

Mean = np = 62 * 0.66 = 40.92

Standard deviaion = sqrt [ n p ( 1 - p) ] = sqrt [ 62 * 0.66 ( 1 - 0.66) ] = 3.730

Using normal approximation,

P(X < x) = P(Z < (X - Mean) / SD )

a)

P(X >= 35) = P(Z > (34.5 - 40.92 ) / 3.73) (With continuity correction )

= P(Z > -1.72)

= P(Z < 1.72)

= 0.9573 (From Z table)

b)

P(X <= 48) = P(Z < (48.5 - 40.92) / 3.73)

= P(Z < 2.03 )

= 0.9788 (From Z table)

c)

X ~ Bin ( n , p)

n = 62 , p = 1 - 0.66 = 0.34

Mean = np = 62 * 0.34 = 21.08

Standard deviaion = sqrt [ n p ( 1 - p) ] = sqrt [ 62 * 0.34 ( 1 - 0.34) ] = 3.730

Using normal approximation,

P(X < x) = P(Z < (X - Mean) / SD )

P(X < 30) = P(Z < (29.5 - 21.08) / 3.73) (With continuity correction )

= P( Z < 2.26)

= 0.9881 (From Z table)

d)

P(X > 20) = P(Z > (20.5 - 21.08) / 3.73) (With continuity correction )

= P(Z > -0.16)

= P(Z < 0.16)

= 0.5636 (From Z table)