LEC 7: CH8:𝜒 2 Goodness-of-Fit Test: Fitting Probability Models to Frequency Data

Example of a probability model: the proportional model

A proportional model is a probability model in which the frequency of occurrences of events is proportional to number of opportunities

Hypotheses for X 2 test

H0 : Observed data come from an expected probability distribution (e.g., from our proportional model)

HA : Observed data do not come from the expected probability distribution

X² Goodness-of-fit test

Uses 𝜒 2 test statistic (Greek letter chi; “kye”)

• Compares observed frequencies against a frequencies expected under null hypothesis

Test statistic for x² test

How to calculate x2

in my own words:

the number of players are the observed, the expected states it clearly. subtract the observed with the expected and divide it by the expected. once you get the values individually for each month, add all of the values to get x² (in this case its 53)

Getting a P-value form  2

the 11 is degrees of freedom

Degrees of freedom for x 2 test

df = (Number of categories) – (Number of parameters estimated from the data) – 1

Degrees of Freedom

specieis which of a family distrubition to use

Critical value

The value of the test statistic where P = alpha

how to read a chi sq table

Assumptions of x^ 2 test

No more than 20% of categories have Expected < 5

• No category with Expected 

Discrete probability models

this is just telling you which test to do based on the categories

Fitting null models: The Poisson distribution

describes the probability certain number of events occur in a block of time or space, when those events happen indepdendtly of each other and occur with = probability at every point in time or space

Example of the Poisson distribution in biology

How can we calculate the Poisson distribution?