Chi-Squared Test and Probability – A Step-by-Step Guide

What is a Chi-Squared Test?

A chi-squared (\(\chi^2\)) test is a statistical method used to analyze tabular data with frequencies or counts. It helps determine whether two qualitative variables are related or independent. Common applications include examining:

Hair colour vs. eye colour
Gender vs. handedness
Social status vs. crime rates

Example Dataset: Handedness by Gender

Here’s a contingency table showing gender and handedness:

	Right-Handed	Left-Handed	Total
Male	43	9	52
Female	44	4	48
Total	87	13	100

Hypotheses:

Null hypothesis (\(H_0\)): Gender and handedness are independent
Alternative hypothesis (\(H_1\)): There is a relationship between gender and handedness

Conducting the Chi-Squared Test

Step 1: State the Hypotheses

\(H_0\): The variables are independent
\(H_1\): The variables are dependent

Step 2: Compute the Test Statistic

Use this formula for each cell:

\[ \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \]

Where:

\(O_{ij}\) = Observed frequency
\(E_{ij}\) = Expected frequency if \(H_0\) were true

Step 3: Calculate and Sum the Terms

\[ T = \frac{(20 - 15)^2}{15} + \frac{(10 - 15)^2}{15} + \cdots = 6.67 \]

Step 4: Compare with the Critical Value

Degrees of freedom: \((2-1)(3-1) = 2\)
Critical value at 5% significance: 7.378

Decision:
Since 6.67 < 7.378, we fail to reject the null hypothesis. There’s insufficient evidence of a relationship between gender and course choice.

Probability Insights

Student Distribution by Course and Gender

Program	Male	Female	Total
Maths	20	10	30
Equine Studies	10	20	30
Chemistry	30	30	60
Total	60	60	120

Basic Probability

Probability a student is in Equine Studies:

\[ P(\text{Equine}) = \frac{30}{120} = 0.25 \]

Conditional Probability

Probability a student is in Equine Studies given they are female:

\[ P(\text{Equine} | \text{Female}) = \frac{20/120}{60/120} = 0.33 \]

Chi-Squared Test Formula (General)

\[ \chi^2 = \sum_{i=1}^{n} \frac{(O_i - E_i)^2}{E_i} \]