Question

Compute​ P(X) using the binomial probability formula. Then determine whether the normal distribution can be used to estimate this probability. If​ so, approximate​ P(X) using the normal distribution and compare the result with the exact probability.

n=60​, p=0.5 and X=37

A. find P(X)

B. find P(x) using normal distribution

C. compare result with exact probability

Answer 1

a)

here this is binomial with parameter n="60 and p=0.5

P(X=37)=

(60C37)0.537(1−0.5)(60-37)    =

0.0203

b)

here mean of distribution=μ=np=	30.00
and standard deviation σ=sqrt(np(1-p))=	3.87
for normal distribution z score =(X-μ)/σx

therefore from normal approximation of binomial distribution and continuity correction:

probability P(X=37) =P(36.5<X<37.5)=P((36.5-30)/3.873)<Z<(37.5-30)/3.873)

=P(1.68<Z<1.94)=0.9738-0.9535=0.0203

c)

both result are approximately the same till 4th decimal places