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I have the following observations

Oberservation ; Count

-1.67 ; 726

18.33 ; 33

148.33 ; 15

This is obviusly not normal distributed :S

How can I make a test for $H_0: \mu = 0$ or even better is it possible to make a confidence interval for the mean?

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What are these values? What's the context? Offhand, the .67 and .33 make me thing something is being divided by 3.... Did this start as a count? My blog post how to ask a statistics question may help –  Peter Flom Dec 14 '13 at 12:42
    
No it is not a count. The numbers are amounts of currency. (To be specific it is in danish crowns and hundredths.) It is the revenue of some online actions. (I did almost write other numbers just to avoid confusion.) –  rlp Dec 14 '13 at 12:47
    
I think you should look into bootstrap. There are packages about this in R, e.g. boot and bootstrap, –  Peter Flom Dec 14 '13 at 12:53
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1 Answer

While the original distribution is clearly non-normal, the sample size is so large that the distribution of the mean will be approximately normal:

enter image description here

(that's the distribution of the sample mean for 10000 samples of the same size as your sample from the empirical cdf. Which is to say, it's the bootstrap distribution of the sample mean).

Further, the distribution of the standard error of the mean is pretty tight, so that you could reasonably treat the null distribution as normal with $\sigma = s$. So you could do a z-test.

Or you could base a test directly off the bootstrap distribution above; it suggests a very small p-value.

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