Question

Question 3 Consider a binomial distribution with a success probability of p = 0.65 and repeated 100 times. What is the normal distribution that approximates this binomial distribution? Use the approximation to find the probability that the number of successes is between 60 and 70 in the 100 repetitions. Next consider the same binomial distribution except that it is repeated 1,000 times. What is the normal distribution that approximates this binomial distribution with 1,000 repetitions? Use the approximation to find the probability that the number of successes is between 600 and 700 in the 1,000 repetitions. Find such that the probability of the number of successes between 650 - x and 650 + in 1,000 repetitions in the same as the probability of success between 60 and 70 in 100 repetitions. In STATA, perform the following: set oba 100 gen B100 = binomial(100,0.65) histogram B100 set obs 1000 gen B1000 = rbinomial (1000,0.65) histogram B1000 What do these commands do! Do the histograms illustrate the results above.

Answer 1

X~binomial(p=0.65)

a)

Y=100*X ~ Normal(mean=100*0.65=65,variance=n*p*(1-p)=100*0.65*0.35=22.75)

Y~ Normal(65,22.75)

b)

Use the approximation to find the probability that the number of successes is between 60 and 70 in the 100 repetitions

p(60 < Y < 70)

=NORMDIST(70,65,SQRT(22.75),1)-NORMDIST(60,65,SQRT(22.75),1)

=0.8527-0.1472
=0.7055

c)

What is the normal distribution that approximates this binomial distribution with 1,000 repetitions

z ~ Normal(650,227.5)

d)

Use the approximation to find the probability that the number of successes is between 600 and 700 in the 1,000 repetitions.

p(60 < Z < 70)

=NORMDIST(700,650,SQRT(227.5),1)-NORMDIST(600,650,SQRT(227.5),1)

=0.999

e)

Find such that the probability of the number of successes between 650 - x and 650 + in 1,000 repetitions in the same as the probability of success between 60 and 70 in 100 repetitions.

we have to find k1 such a that

p(Z< K1)=(1-0.7055)/2

=0.1472

k1=NORMINV(0.1472,650,SQRT(227.5))

=634.1886

k1=650-x

hence x= 650-634.1886

x=15.8114