Chi-Squared Test

Tags

PAS

Cegep/2

Word count

249 words

Reading time

2 minutes

Test to determine whether two qualitative variables are independent

$H_0$: variables are independent
$H_1$: variables are dependent

Steps:

1. Determine the degrees of freedom
2. Construct the table of expected frequencies
3. Compute chi-squared
4. Use probability table to find p-value of chi-squared

Only valid when $n \ge 30$ and $e_{ij} \ge 5 \forall i, j$

Examples

A randomly selected sample was asked to choose their preferred musical genres from a set of choices. Do the results of the sample indicate (at 𝛼 = 5%) whether musical preferences are dependent of the generation to which belongs the respondents?

	Classical/Jazz	Hip Hop/Rap	Pop	Rock/Alternative	Total
Boomers	32	6	11	18	67
Gen X	18	9	23	31	81
Gen Y	11	21	39	17	88
Gen Z	8	31	25	11	75
Total	69	67	98	77	311

$H_0$: the musical preferences are independent of the generation.
$H_1$: the musical preferences are dependent of the generation.

$$\begin{array}{rl}df& =(n-1)(m-1)=(4-1)(4-1)=9\\ {\chi }_9^2& =\sum _{i,j}\frac{(o_{ij}-e_{ij}{)}^2}{e_{ij}}\\ & =\frac{{(32-\frac{67\cdot 69}{311})}^2}{\frac{67\cdot 69}{311}}+\frac{{(6-\frac{67\cdot 67}{311})}^2}{\frac{67\cdot 67}{311}}+\cdots +\frac{{(11-\frac{75\cdot 77}{311})}^2}{\frac{75\cdot 77}{311}}\\ & \approx 70.81\\ p& =P(X_9^2>70.81)\approx 1.05\cdot {10}^{-11}<5\mathrm{\%}\end{array}$$

Therefore, we reject $H_0$ and conclude that preferred musical genres are dependent of the generation.

Contributors

Ajitani Hifumi

Changelog

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