date: 2023-12-30
title: Probability-Theory
status: DONE
author:
  - AllenYGY
tags:
  - Chi-Square
  - Lec5
  - ProbabilityStatics
  - NOTE
created: 2023-12-30T09:21
updated: 2024-04-08T20:13
publish: True

Probability-Theory

The Chi-Square Distribution
Chi-square goodness of fit tests $拟合优度检验$
A chi-square test for independence $独立性检验$

Chi-Square Tests

Test of Multinomial Experiment
- General Test
- Test for homogeneity
- Test for normal distribution
Test for independence

The Chi-Square Distribution

The chi-square distribution depends on the number of degrees of freedom.

There is a family of chi-square distributions.
It is skewed to the right.
It is non-negative.

Test of Multinomial Experiment

H0: multinomial probabilities are p1, p2, … , pk
Ha: at least one of the probabilities differs from p1, p2,…, pk
Choose
Multinomial Experiment
Assumption: The excepted cell frequencies are >5
Critical value is , with k-1 degrees of freedom
Test statistic:

Reject H0 if >

General Test Example

Marital status

Test for homogeneity Example

Support calls

Chi-Square Test for Independence

Chi-square test of independence is used to determine if there is a significant relationship between two qualitative (categorical) variables.

A contingency table is used to investigate whether two traits or characteristics are related.

H0: X and Y are independent
Ha: X and Y are dependent
Assumption: The excepted cell frequencies are > 5
Set the level of significance

Critical value is , with degrees of freedom (r-1)(c-1)
Test statistic:

where:
= observed frequency in cell
= expected frequency in cell
= number of rows
= number of columns
6. Reject H0 if

Chi-Square Test for Independence Example

Summary

Chi-squared test of goodness of fit:

d.f. = k - 1

Chi-squared test of independence in a contingency table:

d.f. = (r - 1)(c - 1)

These formulas and degrees of freedom calculations are fundamental to performing chi-squared tests in statistics.