In a rural area the infected rate of certain disease is 020

In a rural area, the infected rate of certain disease is 0.20 (i.e., 20%). A sample of 20 people is taken from the area - selections are independent. Define X as the number of people in the sample who are infected by the disease.

Are \"normal/possion\" good approximations? Why?

Solution

Here the random variable X is binomial with parameters n=20 and p=0.2

By central limit theorem X can be follows a normal distribution with mean 20*0.2 and variance 20*0.2*0.8

If the sample size is sufficiently large ( that is n tends to infinite) then this approximation holds. But in our problem the sample size is only 20 so there the Normal approximation not holds good.