Hypothesis testing, confidance leves, photon statistics

[captain.joco]

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))/$$\Gamma$$($$\nu$$/2) * (chi^2/2)^($$\nu$$/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

 

[mark726]

!

Your approach is on the right track, but there are a few issues with your calculations.

Firstly, for part a), the number of degrees of freedom is actually 3, not 4. This is because we have 4 observed data points, but we are only estimating 1 parameter (the probability of spin +1). So the chi-squared value would be 3, not 4.

For part b), the number of degrees of freedom is actually 99, not 100. This is because we have 100 observed data points, but we are estimating 1 parameter (the probability of spin +1). So the chi-squared value would be 99, not 100.

Next, you need to calculate the expected number of spin +1 photons for each scenario. For part a), the expected number of spin +1 photons would be (1/2)*4 = 2. For part b), the expected number of spin +1 photons would be (1/2)*100 = 50.

Then, you can calculate the p-value using the chi-squared distribution with the appropriate degrees of freedom. For part a), the p-value would be P(3|3) = 0.223. For part b), the p-value would be P(99|99) = 0.493.

Finally, you can use the p-value to determine the confidence level at which the null hypothesis (equal probability of spin +1 and -1) can be rejected. This can be done using a p-value table or a statistical software. For example, with a p-value of 0.223, the null hypothesis can be rejected at the 77.7% confidence level.

To determine the most likely underlying probability of the source emitting a spin +1 photon, you can use the maximum likelihood method. This involves finding the value of the probability that maximizes the likelihood function, which is given by the product of the probabilities for each observed data point. In this case, the maximum likelihood estimate for both part a) and b) would be 1, as all observed data points have spin +1.

The 68% confidence interval for the underlying probability can be calculated using the Wald method. This involves finding the values of the probability that are within 1 standard deviation of the maximum likelihood estimate. In this case, the 68% confidence interval would be (0.77, 1) for both part a) and