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Lets say that you know the mean and the standard deviation of a regularly distributed dataset. How do you find the probability that a random sample of n datapoints results in a sample mean less than some x?

Example- Lets say the population mean is 12, and the standard deviation is 4, what is the probability that a random sample of 40 datapoints results in a sample mean less than ten?

Yes, this is a homework problem, but I changed the numbers. Go ahead and change them again if you like- I just want to know how to do these kinds of problems. The professor is... less than helpful.

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By regularly-distributed, do you mean normally-distributed? Also, you're basically asking about the sampling distribution of the mean, a.k.a distribution of the sampling mean,… – FBD Sep 2 '13 at 23:10
I don't know, and I don't have one. The question says regularly distributed. I know what normally distributed means, and I think that's what was meant, but the text of the problem said regularly distributed. – IgneusJotunn Sep 2 '13 at 23:11
Also, your link is broken sir. – IgneusJotunn Sep 2 '13 at 23:15
up vote 1 down vote accepted

If you mean "normally distributed", then the distribution of the sample mean is normal with the same expected value as the population mean, namely $12$, and with standard deviation equal to the standard deviation of the population divided by $\sqrt{40}$. Thus it is $4/\sqrt{40}\approx0.6324555\ldots$. The number $10$ deviates from the expected value by $10-12=-2$. If you divide that by the standard deviation of the sample mean, you get $-2/0.6324555\ldots\approx-3.1622\ldots$. That means you're looking at a number about $3.1622$ standard deviations below the mean. You should have a table giving the probabilty of being below number that's a specified number of standard deviations above or below the mean.

If you don't mean normally distributed, then the sample size of $40$ tells us that if the distribution is not too skewed, the distribution of the sample mean will be nearly normally distributed even if the population is not.

The expected value and standard deviation of the sample mean stated above do not depend on whether the population is normally distributed nor even on whether it's highly skewed.

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