Solving Chi-Square Method w/ Poisson Distribution

[Erikve]

Hi,

I trying to complete an exercise, but I have the idea that it is completely wrong what I'm doing. I use Matlab to program the exercise, and that is not the problem.

The exercise is:

The poisson distribution has been used as model for light traffic. That is based on the rationale that the rate of passing cars is constant and the traffic is light such that the cars move independently from each other. The data traffic (see beyond) contains the number of cars passing a crossing in 300 3-minute intervals, counted on various hours of the day and various days of the week. Column one gives the number of cars n, column two the number of intervals with n cars.

0 14
1 30
2 36
3 68
4 44
5 43
6 30
7 14
8 10
9 6
10 5

a. Use the method of moments to fit a Possion distribution to the data.
b. test the goodness of fit using Pearsons chi-square statistic. Calculate the P^*-value
c. Comment on the fit. Give a detailed explanation for the result.

Part a looks like very easy, so that was not a problem to do. I have only no idea what they mean in part b. I have the formula:
X^2=sum((O_i-E_i)^2/E_i with O_i the observed data and E_i the expectation value. But what will be E_i? An element of the possion dist. or something different?

Second question is about the value of p^*. How can I calculate it? I have seen the definition p^*=P(D>=d|model is correct) where D is the difference between the model and the expectation value. But what is d? That should be a value for you think that the method is good... And if I have d and D, I have totally no idea how to calculate P^* :(

Thanks for your answers and your time!

 

[EnumaElish]

E_i can be anything you want; typically 0.

Look at a Chi-sq table; the P* will be stated for the X^2 you have calculated.

 

[Architeuthis Dux]

Hello,

Firstly, it is great that you are using Matlab to program and solve this exercise. It is a powerful tool for scientific analysis and modeling. Now, let's address your questions and concerns about the exercise.

Part a is indeed straightforward, as it asks you to use the method of moments to fit a Poisson distribution to the given data. This involves using the mean and variance of the data to estimate the parameters of the Poisson distribution (lambda). You can refer to any standard statistics textbook or online resources for the specific steps to follow.

Moving on to part b, you are correct in your understanding of the formula for Pearson's chi-square statistic. E_i in this case refers to the expected number of intervals with n cars, based on the fitted Poisson distribution. So for example, for n=0, the expected number of intervals would be E_0 = λ^0 * e^-λ / 0!, and for n=1, E_1 = λ^1 * e^-λ / 1!, and so on. You can calculate these values for each n and then use the formula to calculate the chi-square statistic.

The value of p^* (also known as the p-value) is the probability of obtaining a chi-square statistic as extreme or more extreme than the one calculated, assuming the null hypothesis (in this case, that the Poisson distribution is a good fit for the data) is true. You can use statistical software or tables to calculate this value, or you can use the formula p^* = 1 - F(x^2), where F(x^2) is the cumulative distribution function of the chi-square distribution with degrees of freedom equal to the number of categories minus the number of estimated parameters.

Finally, for part c, you will need to interpret the p-value to determine the goodness of fit. A low p-value (usually less than 0.05) indicates that the Poisson distribution is not a good fit for the data, while a high p-value suggests that the data is consistent with the Poisson distribution. This interpretation is based on the assumption that the null hypothesis is true.

I hope this helps clarify your questions and guides you in completing the exercise. Remember, don't hesitate to consult additional resources or seek help from your instructor or colleagues if needed. Good luck!