Solved – If the parent distribution is skewed, but the sample size is greater than 30, is the distribution of the sample mean nearly Normal

distributionsnormal distributionsample

I have my AP Stats exam tomorrow and had this question.

I apologize if it is too simplistic for this forum.

I'm pretty sure the sample's distribution is nearly normal, but not 100%, can someone enlighten me?

Also, I'm sure this question must be a duplicate, but I can't find anything about this specifically, so please help me/direct me to a [potential] duplicate.

Thank you!

Best Answer

The answer to the question in your title is "no" (sometimes it would be the case - if you pin down what you mean by nearly and define the circumstances, it may happen - but the number 30 is essentially irrelevant in general; it's no more special than 10 or 100 or 100000).

For an example, consider a skewed parent distribution - say the gamma distribution with shape parameter 0.01 and whatever (common) scale you like. The mean of 30 of them is also-gamma now with shape 0.3 (and a different scale). It's still quite skew and clearly non-normal. At n=100, you get an exponential distribution for the mean -- still pretty skew, still non-normal. At n=3000 ... it's starting to look pretty reasonable and for some definitions of "nearly" would be fine (but if you're trying to work out extreme tail probabilities, it may still not be enough).

[The mere fact that they mention n=30 has me concerned that they expect an answer which will be wrong. Check your materials/notes/text to be sure that you know what answer they expect to get, irrespective of the actual situation (do I bemoan the need to give this advice? Why, yes, indeed I do).]