9.3.2 Chi-Square Goodness-of-Fit Test

Question 1

What is a goodness-of-fit test?

Answer 1

Testing whether your assumptions about the distribution is valid or not.

e.g. test whether a specified distribution or model fits a dataset.

Question 2

Say you have some sample data {x1, x2, …, xn} and want to test the assumption of Poisson distribution. What are the steps?

Answer 2

1. Estimate λ using the data

2. Goodness of fit test to check if this is correct.

Question 3

What is the basic idea of g.o.f?

Answer 3

• Partitition all possible values into c categories.
• Treat each category as a possible outcome of a multinomial distribution.
• Use chi-square with appropriate d.f. to look at the probability of seeing whatever outcome you got.

Question 4

Ex. g.o.f:

During WWII, a 36km^2 area of South London was divided into 576 small squares of 0.25km^2 each to record bomb hits. The data are as follows:

hits: # squares

0: 229
1: 211
2: 93
3: 35
4: 7
5: 0
6: 0
7: 1

Test whether a Poisson model would fit the data at α = 0.05

Answer 4

-Find expected value of this sample data, λ:

Σ (# hits) (# squares) / (total #)

Ho: # hits per .25 km^2 follows Poisson distribution.

Ha: # hits does not follow Poisson dist. (at least one Pi != Pio)

Combine last columns together until there is a frequency for that column greater than 5

Question 5

Given X ~ N(0,1), what is chi-square?

Answer 5

Q = X^2
Q ~ X^2_1

Or, Q is distributed among the chi-square distribution with 1 d.f

Question 6

What does it mean for a chi-square r.v. to have 2 d.f?

Answer 6

You have 2 r.v’s.
Sample X1 from a normal distribution, square it, and do the same with X2.

Q2 = X1^2 + X2^2
means Q ~ X2^2