Question

Using Central Limit Theorem to establish the following result: √ n(ˆp−p) ·∼ N(0, p(1− p)) for large n.

Answer 1

Central Limit therome ( CLT) : For large sample size ( n ), the sampling distribution of the sample mean follows approximately normally distributed with mean is E( mean ) and variance is Varmean).

Here we need to find the mean and variance of

Where is the sample proportion

We know that the sample proportion is an unbiased estimator of the population proportion.

Therefore ,  

And

Therefore ~ N ( *p , p(1-p) )

this implies that - P~ N ( 0 , p(1-p) )

( - p) ~ N ( 0 , p(1-p) )