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 equals = 81​, p equals = 0.82 0.82​, and x equals = 72

Answer 1

Using binomial probability:

for, n = 81 , p = 0.82 and x = 72

P[X=x] =

= 0.0322

Now, for Binomial (n,p) :

E(X) = np = 81 * 0.82 = 66.42

Var(X) = np(1-p) = 81*0.82*0.18 = 11.96

Using normal approximation : As n is 81, quite large, we can use normal approximation.

  

Now, P[X=72] = P[X<73]-P[X<72]

= P[Z<1.90] - P[Z<1.61]

= 0.9713 - 0.9463

= 0.025

So we can see that normal approximation under estimate the proability a bit, But it is still quite good approximation.

If you have queries, ask me below, Thank you