# Given mean and standard deviation, find the probability

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, e.khanacademy.org/math/probability/statistics-inferential/… – 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

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.