Hi,
Let me understand this correctly. You are referring to Exercise 7, which is scan of hypothesis tests to invert it, i.e. to find the parameter value corresponding at a given p-value (e.g. 0.05).
This is done for computing a list and not a significance. In this case the null hypothesis is the S+B model, because we want to compute a limit, i.e. reject the hypothesis that we have signal in our data.
What I was saying in the previous posts refers to a significance test (e.g. Exercise 6).
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Now in a Hypothesis inversion scan (for computing a limit/interval), if we have CLb < CLs+b, it means that the data prefer S+B model. So in this case you cannot really set a limit, but compute an interval. The correct way to do in that case would be to use also a two sided test-statistics and compute a Feldman-Cousins interval.
There is no need to use CLs, but to use directly CLs+b
More information is from slides #76 of
indico.desy.de/getFile.py/acces … nfId=11244 -
The data set used in Exercise 7 is more background-like. You see that CLs+b is always smaller than CLb, apart for very small values of the parameter of interest (signal rate). We see that for s > 6 the CL(s+b) is smaller than 0.05, so we can reject the S+B null hypothesis and set a 95% limit for s at around 6.
Lorenzo