Methods and formulas for Goodness-of-Fit Test for Poisson

Formula

The mean for the Poisson distribution is estimated as:

Computation

Data	2 2 3 3 2 4 4 2 1 1 1 4 4 3 0 4 3 2 3 3 4 1 3 1 4 3 2 2 1 2 0 2 3 2 3

Category (i)	Observed (Oi)	Estimated mean	Poisson probability (pi)
0	2	0 * 2 = 0	p0 = e-2.4 = 0.090718
1	6	1 * 6 = 6	p1 = e-2.4 * 2.4 = 0.217723
2	10	2 * 10 = 20	p2 = e-2.4 * (2.4)2/ 2! = 0.261268
3	10	3 * 10 = 30	p3 = e-2.4 * (2.4)3/ 3! = 0.209014
	7	4 * 7 = 28	p4 = 1 - (p0 + p1 +p2 + p3) = 0.221267

N = 35

Σ (i * O_i) = 84

Estimated Mean =

Notation

Minitab determines categories using the following iterative methods:

Defining the first category

Let p_i = P(X x_i )

Let i = 1: if N*p_i 2, then the first category is defined as " x ₁". If N*p_i < 2, then increase i by one and repeat: if N*p ₂ 2, then the first category is defined as " x ₂". If N*p_i < 2, increase i by one and repeat until N*p_i 2. Stop iterations either when this condition is first satisfied, or when x_i is the third largest data value, and define the first category as " x_i ". If the value of the first category is zero, then the first category is defined as "0" without the "less than or equal to" sign. The probability and expected value associated with the first category is p_i and N*p_i respectively. The observed value for the first category is the number of all data values x_i .

Defining the last category

Conceptually, defining the last category is similar to defining the first category, but Minitab works backward starting from the largest data value.

The last category is " x_j ", where x_j is the largest data value greater than (1 + the data value of the first category), such that the category has an expected value greater than 2. The probability and expected value for the last category are p_j and N*p_j respectively, and the observed value is the number of data values x_j .

Defining the middle categories

After determining the first and the last categories, Minitab determines the categories between them. Let "X k" be the first category, and "X m" is the last category. If all integers between (k, m) have expected values 2, then they all constitute a middle category. If not, Minitab uses a recursive loop to group multiple adjacent integers into categories with expected values 2. There are some situations, such as a data set with few observations, where the expected value of a category will be less than 2.

Notation

Formula

The Poisson probability of the i ^th category (i < k) is,

The Poisson probability for the last category, where i = k,

p_i = 1 – (p₀ + p₁ + ...+ p_k-1)

Notation

Formula

The expected number of observations in i ^th category is N * p_i .

Notation

Formula

Contribution of the I^th category to the chi-square value is calculated as

Notation

Formula

The chi-square goodness-of-fit test statistic is calculated as,

Notation

The p-value is:

Prob (X > Test statistic)

where X follows a chi-square distribution with k - 1 degrees of freedom if you use the MEAN subcommand, or k- 2 degrees of freedom if you do not use the MEAN subcommand.

Computation

2  2  3  3  2  4  4  2  1  1  1  4  4  3  0  4  3  2 
3  3  4  1  3  1  4  3  2  2  1  2  0  2  3  2  3

Category (i)	Observed (Oi)	Estimated mean	Poisson probability (pi)
0	2	0 * 2 = 0	p0 = e -2.4 = 0.090718
1	6	1 * 6 = 6	p1 = e -2.4 * 2.4 = 0.217723
2	10	2 * 10 = 20	p2 = e -2.4 * (2.4)2/ 2! = 0.261268
3	10	3 * 10 = 30	p3 = e -2.4 * (2.4)3/ 3! = 0.209014
	7	4 * 7 = 28	p4 = 1 - (p0 + p1 +p2 + p3 ) = 0.221267

= ( 0.434920 + 0.344527 + 0.080058 + 0.985114 + 0.071545) = 1.91622

k = 5= the number of categories

DF = 5- 2 = 3

p-value = P (X > 1.91622) = 0.590

Notation

Term	Description
k	the number of categories
Oi	the observed number of observations in the ith category.
Ei	the expected number of observations in the ith category.
	chi-square goodness-of-fit test statistic
DF	degrees of freedom