Question

3. The EPA estimates that 63% of the commuters in the city of Los Angeles drive to and from work alone. Suppose 120 Los Angeles commuters are selected at random.

1. (a) Find the probability that in the sample of 120, exactly 75 commuters drive to and from work alone.

2. Use your calculator to find the exact probability that in the sample of 120 more than 75 commuters drive to and from work alone.

3. Find the probability that in the sample of 120 more than 75 commuters drive to and from work alone.

4. Use the Normal approximation to the Binomial distribution to approximate the probability that in the sample of 120 more than 75 commuters drive to and from work alone.

Answer 1

Solution :

here we use binomial distribution with parameter n=120 and p=63% =0.63 and for

Binomial distribution ,P(X=r)=nCrpr(1-p)n-r  

(a)

P(X=75) =0.0746 ( using MS-excel command =BINOMDIST(75,120,0.63,0) )

(b)

P(X>75)=1-P(X<=75)=1-0.4892=0.5108  

P(X<=75)=0.4892 ( using MS-excel command =BINOMDIST(75,120,0.63,1) )

(c)

here mean =n*p=120*0.63=75.6 and

standard deviation=sd=sqrt(n*p*(1-p))=sqrt(120*0.63*(1-0.63))=sqrt(27.972) = 5.2889

for x=75,z=(x-mean)/sd=(75-75.6)/5.2889= -0.1135

P(x>75)=P(z>-0.1135)=1-P(z<-0.1135)=1-0.4548 = 0.5452

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