How to prove statistically that mean jumps when a parameter crosses an unknown threshold value? We have a random numbers generator (it is actually a physical device standing outside of our building). By collecting of about 3000 numbers, generated by the device, we came to the conclusion that the mean of the generated numbers is positive (although it is close to zero). The statistical significance of this conclusion is quite good.
Recently we started to suspect that when temperatures outside are high, the device tends to generate slightly smaller numbers. Even more, we suspect that when temperature is above a threshold temperature, the mean of the generated numbers is negative (although we do not know what is exactly the value of this threshold temperature).
I also need to add, that we have a reason to believe, that the dependency of the mean of the generated numbers on the temperature is a step function. To make it more clear, we believe that when temperature is high enough two parts of the device disconnect from each other and the device switches to a new "regime" (associated with the generation of presumably smaller numbers).
So, my question is, how can we calculate the statistical significance of our hypothesis that there is a temperature triggered switch in the mean of the generated numbers?
We could, for example, find a value of the threshold temperature empirically such that the mean of the numbers, generated by the device, is negative and as low as possible and then we could calculate the statistical significance that the mean of these values is really negative (in the statistical sense). However, I guess that this approach is false, since we have manually chosen the split such that numbers corresponding to the high temperature are small, so it is not fair to prove if they have statistically significant negative mean.
In other words, would a threshold value of the temperature be given, we could just check if we have a statistical evidence to believe that values, corresponding to the high temperatures, have negative mean.
ADDED:
My question is not about physics. The "physical device" does not exists. I made the example just to make it less abstract and easier to grasp.

Your question involves two tasks, estimation and significance testing.
In both cases you have uncertainties about the model (like no specification for the distribution of the "random number generator").
A way to resolve this is to use general methods. These methods work for different types of distributions, but they may not be the most efficient. (For instance, knowing more about the problem might make you choose a better cost function for training the model during fitting, such that the parameter estimates have with higher probability a smaller error)
In your case you could use isotonic regression or regression trees (restricted to a single split) to estimate the parameters and bootstrapping to determine significance.

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*Estimation of parameters
You can use the least squares method. Fit the model such that the square of the residuals is minimised. This method is consistent: you can make your estimate as precise as desired by adding more data.
More specifically your model can be fitted with isotonic regression or regression trees with a square loss function, and that are restricted to having a single split. For this there are many software libraries available that can help you to perform the regression.


*Significance
If you do not know the probability distribution of the observations, then you can not exactly compute the significance. This is because you have no information about the theoretical distribution for the performance of the estimates.
One method would be to use ANOVA (Analysis of variance) to compare the difference between the null model (there's no jump and the mean is independent of temperature) and the alternative model (the mean jumps above some threshold temperature).
If instead of not knowing the distribution, you would know that the distribution of the observations is Gaussian, then you could express the significance by approximating the result of the ANOVA as F-distributed. On the other hand when the distribution is not Gaussian, then possibly you might use the Gaussian distribution as an estimate, or if you suspect a large discrepancy in the distribution then you could use bootstrapping to estimate the distribution.
