# Normal distribution and Null hypothesis

## Main Question or Discussion Point

I have done 3 experiments. For each one of them, I have repeated the same experiment 100 times. Which gives me three sets of 100 numbers.
Experiment 1: for number 30 ---> 100 results
Experiment 2: for number 40 ---> 100 results
Experiment 3: for number 50 ---> 100 results
Basically, I had done other experiments for numbers below 30 and for numbers above 50.
All 100 results for each experiment corresponding to numbers BELOW 30 were negative, and all 100 results for each experiment corresponding to numbers ABOVE 50 were positive.
My results looked like this:
Experiment for number 10: [ -500 -480 -530 -503 -460....]
Experiment for number 20: [ -10 -20 -2 -15 -20....]
Experiment for number 30: [ -5 10 -18 0 -7....]
Experiment for number 40: [ 60 -16 100 -4 200....]
Experiment for number 50: [ -150 850 600 -20 560....]

Experiment for number 60: [ 1560 2000 3500 2200 3100....]
Experiment for number 70: [ 4000 5580 5800 7600 6000....]

The thing is when I do the experiments corresponding to numbers 30, 40 and 50, each set of 100 results contains both positive and negative numbers (but smaller numbers).
I am trying to see which one (or what range) between the numbers 30, 40 or 50 is considered as the 'break even' point (I mean the value where the 100 results are considered to be mostly 0).
As of now, the only thing I can say, is that it appears that the break even point is a value between 20 and 60 ( at 20, all 100 results are negative, at 60, all 100 results are positive). But I would like a little bit more precision.
How do I go about doing this? I am thinking about maybe testing the results for number 30, and see whether these follow a Normal distribution with mean 0. Then do the same with 40 and 50. I am not quite sure whether this is the correct way to do this though.
Someone suggested to use Null and alternative Hypothesis:
$H_0 = X$~ $N (\mu, \sigma ),$ where $\mu < 0$
$H_A \neq X$~ $N (\mu, \sigma ).$
But I don't really know whether that is the right way to go, how to do this or even what conclusion I can get from it.
Any help is very appreciated.

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You could cook up a model for how you expect the mean of your results to vary with this "number". For instance, if you were happy with a linear fit through those central three data points (more would be better), then you could constrain

mean = A * number + B

and then you could use your normal model of the scatter for each "number" to extract some confidence intervals on A and B, and then on the "number" for which mean=0. Of course if you have a proper physical model that would be much better.

mfb
Mentor
I am thinking about maybe testing the results for number 30, and see whether these follow a Normal distribution with mean 0
That is unlikely. Look if the results follow any normal distribution, and find its mean and standard deviation. Is it positive? If yes, significantly?

You can do that for all sets (including 10, 20 and so on) and try to find a relation between the means and the tested number. That will allow to find the zero-crossing with a smaller uncertainty.