Let's say you were given a small sample (n = 15 or so), and you knew the sample mean, and sample variance, but you did not know the distribution.

Would you use a z-test or a t-test?

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- Jan 9th 2010, 07:49 AMstatmajorHypothesis Testing
Let's say you were given a small sample (n = 15 or so), and you knew the sample mean, and sample variance, but you did not know the distribution.

Would you use a z-test or a t-test? - Jan 9th 2010, 10:33 AMANDS!
Using the t-distribution is only allowed if we know the population that the sample came from is approx. normal. So regardless of whether you knew all the stats, unless the distribution of your population was approx. normal, you wouldn't be able to use the t-distribution.

- Jan 9th 2010, 10:38 AMstatmajor
So which test word I use for hypothesis testing?

- Jan 9th 2010, 10:51 AMANDS!
With a sample size that small, your only option is to use the t-distribution (and applicable tests). However, unless you know the underlying distribution of the population, you can't do anything with it.

Remember the central limit theorem is based on the fact that with a large enough sample size (30 or more), no matter what the underlying population distribution, the population of samples means from samples sizes n=(30 or more) will be normally distributed.

The reason the t-distribution works, is because we know the distribution of the population underneath, so whether we have 15 values or 5, those values will still behave as if pulled from a normal population.

Can you write the actual problem you have? - Jan 9th 2010, 10:55 AMstatmajor
I'm not working on a problem. I was just curious to what I would in that situation if it ever arose.

So same problem, except n = 41. So according the CLT, since n is large enough (41>30), the population distribution is approximately normal, I could then use the t-distribution or the normal distribution?

And if the population distribution was a normal distribution (the problem in my first post), I could either use the t-distribution or the normal distribution? - Jan 9th 2010, 11:10 AMANDS!Quote:

Originally Posted by**statsmajor**

THAT is what the Central Limit Theorem is about, and what allows us to do hypothesis testing: regardless of the underlying population, what are the chances that if I draw a sample of size "n", that all the values of that sample will give me a mean of some number that differs from the stated mean of my population. So the underlying population could be exponential, it could be uniform, it could be some crazy equation f(x)=e^-e*x-sin(e^x), or whatever - doesn't matter. Large enough population, and you're good to go.

In your case, your sample size is less than 15. So you would have to know the distribution of the population to construct any hypothesis testing.

In your reworked problem if you had a sample size of 41, you could use either a z-test, or a t-test (although a t-test would be slightly pointless as you have population statistics - mean, variance and std). - Jan 9th 2010, 11:13 AMstatmajor
Thanks for clearing that up for me.

- Jan 12th 2010, 07:44 AMnovice
Statmajor,

For small samples (N<6), Student-t is perfect because it's built in with a correction factor, but there are some other methods you will find interesting. Check out the U-test and H-test. They are supprising easy. For testing, I found the non-parametric tests are more practicle. Try non-parametric tests, and you will have a lot of fun. It's like playing rather than toiling. If I were an actuary, I would do non-parametric test before I do serious number crunching. These are usefull for determining whether your sample is random or not. If your sample is not randomly distributed, why bother? - Jan 14th 2010, 07:22 AMnovice
Statmajor,

Sorry for the wrong information I gave you. I just realized that I didn't talk at the same wavelength.