Hypothesis testing, confidance leves, photon statistics

  • #1

Homework Statement


A source is supposed to emit photons with spin +/-1 independently at equal rate.
a) after measuring 4 photons, all have spin +1;
b) after measuring 100 photons, 60 have spin +1.
at which confidence level can the hypothesis be rejected.


Calculate for each of the outcomes the most likely underlying probability of the source emiting a photon of spin 1. What is the 68% confidence interval on each of the probabilities? Which underlying distribution has been assumed to estimate the 68% confidence region.

Homework Equations



chi-squared = (observed - expected ) ^2 /expected, where I expect on average the spin to be 0

p(chi-squared|degrees of freedom) = (e^(-chi^2/2))/[tex]\Gamma[/tex]([tex]\nu[/tex]/2) * (chi^2/2)^([tex]\nu[/tex]/2-1)


The Attempt at a Solution


I have proceeded to calculate chi-squared parameter. For part a) chi-sqared is 4, and for part b) is 20. The number of free parameters for part a is 4, as i have 4 data points, and no free parameters, and for part b, number of degrees of freedom is 100.

Then I calculate P(4|4) = integral over P(chi-squared) between 4 and infinity gives 0.406.


but for part b I have number of degrees of freedom to be 100.. and the arithmetics becomes complicated. Is this anywhere near correct?


I don't know the last few parts... Any help? Thank you in advance
 

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