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I am reading about the familiar hypothesis test for proportions, using the normal approximation for large sample sizes. Some people say (write) that the condition for using the approximation is np>5 and nq>5. Others say np>10 and nq>10. Which one of these two is correct and why ? Thank you.

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Neither is correct, and neither is incorrect; indeed there are other rules besides these (some of a different form). The second criterion is stricter than the first.

You're talking about approximation; there's no right or wrong answer unless you specify a lot more. For one person in one circumstance some particular rule of thumb will be fine, and for another person or another circumstance it won't guarantee that some calculation based on it will be sufficiently close for their purposes.

Computers can calculate binomial probabilities with high accuracy and are not bothered by a little tedium -- why would do you need a normal approximation at all?

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