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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

That guideline for the normal approximation to the binomial is based on mathematical statistics results that show how fast the binomial converges to the normal. The most accurate way to figure it out would be to calculate or simulate sampling from your population and seeing how the sample average behaves. Absent that, theoretical statisticians have derived a number of criteria, none of them foolproof or guaranteed, that indicate when you will not be "too wrong" in using a Normal distribution with the parameters you indicate.

For you second question: You don't "know," it's an assumption...or, in some cases, a hope ;-) In actual statistics, you really should verify distribution assumptions, not just assume it. However, homework problems typically just assert these things to make the problems easier to solve. As you get experience with different datasets, you may come to know when a normal approximation is OK..but you are never guaranteed of a normal distribution unless you know your population is normal.

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