Let x be a binomial random variable with n100 and p8 The pro

Let x be a binomial random variable with n=100 and p=.8. The probability of less than 78successes, when using the normal approximation for binomial is ________

a) 0.2659

b) 0.5

c) 0.4156

d) 0.0002

e) 0.64

f) 0.04

Solution

mean=n*p=100*0.8=80

standard deviation =sqrt(n*p*(1-p))

=sqrt(100*0.8*0.2)=4

So probability of less than 78successes, when using the normal approximation for binomial is

P(X<78) =P(X<77.5) (by using continuous correction)

= P((X-mean)/s <(77.5-80)/4)

=P(Z<-0.625)

=0.2659 (from standard normal table)

Answer: a) 0.2659