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 60% of people who are murdered actually knew the person who committed the murder.† Suppose that a detective file in New Orleans has 58 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

(a)

n = 58

p = Probability of knowing their murderers = 0.60

q = 1 - p = 0.40

np = 58 X 0.60 = 34.8 10

nq= 58 X 0.40 = 23.2 10

Both conditions are satisfied. So, it is appropriate to use the normal approximation to the binomial.

= np = 58 X 0.60 = 34.8

To find P(X 35):
Applying Continuity Correction, we get:

To find P(X > 34.5):

Z = (34.5 - 34.8)/3.7310

= - 0.0804

By Tecnology, Cumulative Area Under Standard Normal Curve = 0.4680

So,

P(X > 34.5):= 1 - 0.4680 = 0.5320

So,

Answer is:

0.5320

(b)

To find P( X 48):

Applying Continuity Correction, we get:

To find P(X < 48.5)

Z = (48.5 - 34.8)/3.7310

= 3.6719

By Tecnology, Cumulative Area Under Standard Normal Curve = 0.0.9999

So,

Answer is:

0.9999

(c)

fewer than 30 victims did not know their murderers is equivalent to greater than or equal to 30 victims knew their murderers.

To find P(X 30):

Applying Continuity Correction, we get:

To find P(X > 29.5):

Z = (29.5 - 34.8)/3.7310

= - 1.4205

By Tecnology, Cumulative Area Under Standard Normal Curve = 0.0777

So,

P(X > 29.5):= 1 - 0.0777 = 0.9223

So,

Answer is:

0.9223

(d)

More than 20 victims did not know their murderers is equivalent to less than or equal to 20 victims knew their murderers.

To find P(X 20):

Applying Continuity Correction, we get:

To find P(X < 20.5):

Z = (20.5 - 34.8)/3.7310

= - 3.8328

By Tecnology, Cumulative Area Under Standard Normal Curve = 0.0001

So,

Answer is:

0.0001