1 . the following problem, check that it is appropriate to use the normal approx

Question

1 . 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 63 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

2 .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.

Do you take the free samples offered in supermarkets? About 61% of all customers will take free samples. Furthermore, of those who take the free samples, about 37% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 321 customers passed by your counter. (Round your answers to four decimal places.) (a) What is the probability that more than 180 will take your free sample?


(b) What is the probability that fewer than 200 will take your free sample?


(c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability

P(buy|sample) = 0.37, while P(sample) = 0.61.


(d) What is the probability that between 60 and 80 customers will take the free

Explanation / Answer

1. Here n=63 and p=0.66

So np=41.58>=10 also npq=14.14>10 so we can use approximation to normal

Now mean=np=41.58 and sd=sqrt(npq)=3.76

a. P(x>=35)=P(z>=35-41.58/3.76)=P(z>=-1.75)=0.5-P(0<z<-1.75)=0.5+0.4599=0.9599

b. P(x<=48)=P(z<=48-41.58/3.76)=P(z<=1.71)=0.5+P(0<z<1.71)=0.5+04564=0.9564

c. P(x<30)=P(z<-3.1)=0.5+P(0<z<-3.1)=0.5-0.499=0.001

d. P(x>20)=P(z>20-41.58/3.76)=P(z>-5.7)=0.5-P(0<z<-5.7)=0.5+0.5=1

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