Answer to Question #204786 in Statistics and Probability for sam21ucd

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.