Question

Suppose Y is a random variable that follows a binomial distribution with n = 25 and π = 0.4. (a) Compute the exact binomial probability P(8 < Y < 14) and the normal approximation to this probability without using a continuity correction. Comment on the accuracy of this approximation. (b) Apply a continuity correction to the approximation in part (a). Comment on whether this seemed to improve the approximation.

Answer 1

a)

frm binomial distribution: P(8<Y<14)= =0.6487

mean =np=25*0.4=10

and std deviaiton =sqrt(np(1-p))=2.45

therefor from normal approximation:

P(8<Y<14)=P((14-10)/2.45<Z<(8-10)/2.45)=P(-0.82<Z<1.63)=0.9484-0.2061=0.7423

error in approximation=|0.6487-0.7423|=0.0936

here the estimate is not accurate as the difference is singificantly high

b)

from continuity correction

P(8<Y<14)=P((13.5-10)/2.45<Z<(8.5-10)/2.45)=P(-0.61<Z<1.43)=0.9236-0.2709=0.6527

error in approximation=|0.6487-0.6527|=0.0040

this seems to be quite accurate as diffeence in approximation and actual is very low