# Sampling and normal distributions

There's a criteria used in my book which describes that $\hat{p}$ is approximately normal provided that $npq \geq 10$. Where did this assumption come from? $n$ is the sample size, $\hat{p} = \frac{x}{n}$, p is the population proportion and $q=1-p$.

My other question deals with normal distribution. Lets say there's a doctor who is measuring 200 three old patients. He finds that $\mu = 38.72$ inches and $\sigma = 3.17$ inches. How do you know that the area under the normal curve between 35 and 38 inches is equal to the area of the standard normal curve between the $z$ scores corresponding to the heights of 35 and 38 inches?

In other words, the standardized version of $x=35$ is -1.17 and the standardized version of $x=38$ is -0.23. How do you know that $P(35 \leq x \leq 38) = P(-1.17 \leq z \leq -0.23)$ ?

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It would be helpful to know what $n$, $p$, $\hat{p}$ and $q$ are. –  Nameless Oct 27 '13 at 22:32