Question

Find the probability of more than 30 heads in 50 flips of a fair coin by using the normal approximation to the binomial distribution.
a) Check the possibility to meet the requirements to use normal approximation (show your calculation)

b) Find the normal parameters of the mean(Mu) and standard deviation from the binomial distribution.

c) Apply normal approximation by using P(X>30.5) with continuity correction and find the probability from the table of standard normal distribution.

Answer 1

n = 50, p = 0.5

a)

the normal approximation works best when p is close to 0.5 and it becomes better and better when we have a larger sample size n. This can be summarized in a way that the normal approximation is reasonable if both and as well.

as p = 0.5 which is close to 0.5

and n*p = 50*0.5 = 25 >=10

and n*(1-p) = 50*(1-0.5) = 25 >= 10

as all the requirements are satisfied normal approximation will be a good estimate for binomial

b)

mu = n*p = 50*0.5 = 25 ( mean of binomial distribution)

sigma2 = n*p*(1-p) = 50*0.5*0.5 = 12.5 ( variance of binomial distribution )

sigma = (sigma2)1/2 = (12.5)1/2  = 3.535534

c)

= 0.05989746