Question

Explain why the sampling distribution of the sample proportion is assumed to be Normal. Show your steps and write down any assumptions you make.

Answer 1

By Central Limit Theorem:

$\frac{\overline{x}-\mu}{\sigma/\sqrt{n}}\to N(0,1)$ for $n\to \infin$

That is why the sampling distribution of the sample proportion is assumed to be Normal.

The standard deviation of sample proportion:

$\sigma=\sqrt{\frac{p(1-p)}{n}}$

Since the sample size n appears in the denominator of the square root, the standard deviation does decrease as sample size increases. Finally, the shape of the distribution of p-hat will be approximately normal as long as the sample size n is large enough.