UNIT 5 • TESTING OF HYPOTHESIS

Chi-Square (χ²) Test
Goodness of Fit Test

The non-parametric Chi-Square (χ²) Test evaluates whether observed counts across categorical bins match expected theoretical frequencies, testing if discrepancies are statistically significant!

CHI-SQUARE GOODNESS OF FIT STATISTICRIGHT-TAILED DIST

\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i} \quad \text{with } df = k - 1

1. Residual ComponentNormalized squared difference:

\frac{(O_i - E_i)^2}{E_i}

2. Degrees of FreedomNumber of categories minus 1:

df = k - 1

3. Critical DecisionReject

H_0

\chi^2_{calc} > \chi^2_{\alpha, df}

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Test for equal distribution across 4 categories:

O = [25, 19, 16, 20], \quad E_i = 20 \quad (\text{Equal expected counts})

Test $H_0$ at significance level $\alpha = 0.05$ !

STEP 01 • TABULATE CATEGORY RESIDUAL COMPONENTS

Category (i)	Observed (O_i)	Expected (E_i)	O_i - E_i	(O_i - E_i)² / E_i
1	25	20	+5	1.25
2	19	20	-1	0.05
3	16	20	-4	0.80
4	20	20	0	0.00
Total	80	80	0	χ²_calc = 2.10

STEP 02 • CHI-SQUARE CRITICAL COMPARISON

Compare test statistic with critical threshold for $df = 4 - 1 = 3$ :

\chi^2_{\text{calc}} = \sum_{i=1}^4 \frac{(O_i - E_i)^2}{E_i}

\chi^2_{\text{calc}} = \frac{(25-20)^2}{20} + \frac{(19-20)^2}{20} + \frac{(16-20)^2}{20} + \frac{(20-20)^2}{20}

\chi^2_{\text{calc}} = 1.25000 + 0.05000 + 0.80000 + 0.00000 = 2.10000

\chi^2_{\text{calc}} = 2.10000 < \chi^2_{0.05, 3} = 7.815

FINAL RESULT

Decision: Fail to Reject H₀ (Observed Counts Fit Expected Model)

\boxed{\text{Fail to Reject } H_0: \chi^2_{\text{calc}} = 2.10 < 7.815}

Because calculated $\chi^2_{calc} = 2.10$ is well below critical value $7.815$ , we fail to reject the null hypothesis—concluding that observed frequencies match the theoretical distribution!

LABORATORY

Interactive Live Calculator & Visualizer

GOODNESS OF FIT LABORATORY

Chi-Square Test Engine ( $\chi^2$ )

PreviousStudent's t-Test

Unit 5•27 of 28

NextF-Test

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 5 • TESTING OF HYPOTHESIS

Chi-Square (χ²) Test
Goodness of Fit Test

The non-parametric Chi-Square (χ²) Test evaluates whether observed counts across categorical bins match expected theoretical frequencies, testing if discrepancies are statistically significant!

CHI-SQUARE GOODNESS OF FIT STATISTICRIGHT-TAILED DIST

\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i} \quad \text{with } df = k - 1

1. Residual ComponentNormalized squared difference:

\frac{(O_i - E_i)^2}{E_i}

2. Degrees of FreedomNumber of categories minus 1:

df = k - 1

3. Critical DecisionReject

H_0

\chi^2_{calc} > \chi^2_{\alpha, df}

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Test for equal distribution across 4 categories:

O = [25, 19, 16, 20], \quad E_i = 20 \quad (\text{Equal expected counts})

Test $H_0$ at significance level $\alpha = 0.05$ !

STEP 01 • TABULATE CATEGORY RESIDUAL COMPONENTS

Category (i)	Observed (O_i)	Expected (E_i)	O_i - E_i	(O_i - E_i)² / E_i
1	25	20	+5	1.25
2	19	20	-1	0.05
3	16	20	-4	0.80
4	20	20	0	0.00
Total	80	80	0	χ²_calc = 2.10

STEP 02 • CHI-SQUARE CRITICAL COMPARISON

Compare test statistic with critical threshold for $df = 4 - 1 = 3$ :

\chi^2_{\text{calc}} = \sum_{i=1}^4 \frac{(O_i - E_i)^2}{E_i}

\chi^2_{\text{calc}} = \frac{(25-20)^2}{20} + \frac{(19-20)^2}{20} + \frac{(16-20)^2}{20} + \frac{(20-20)^2}{20}

\chi^2_{\text{calc}} = 1.25000 + 0.05000 + 0.80000 + 0.00000 = 2.10000

\chi^2_{\text{calc}} = 2.10000 < \chi^2_{0.05, 3} = 7.815

FINAL RESULT

Decision: Fail to Reject H₀ (Observed Counts Fit Expected Model)

\boxed{\text{Fail to Reject } H_0: \chi^2_{\text{calc}} = 2.10 < 7.815}

Because calculated $\chi^2_{calc} = 2.10$ is well below critical value $7.815$ , we fail to reject the null hypothesis—concluding that observed frequencies match the theoretical distribution!

LABORATORY

Interactive Live Calculator & Visualizer

GOODNESS OF FIT LABORATORY

Chi-Square Test Engine ( $\chi^2$ )

PreviousStudent's t-Test

Unit 5•27 of 28

NextF-Test

Chi-Square (χ²) Test Goodness of Fit Test

Step-by-Step Worked Example

Decision: Fail to Reject H₀ (Observed Counts Fit Expected Model)

Interactive Live Calculator & Visualizer

Chi-Square Test Engine (χ2\chi^2χ2)

NETZ Engine

Chi-Square (χ²) Test Goodness of Fit Test

Step-by-Step Worked Example

Decision: Fail to Reject H₀ (Observed Counts Fit Expected Model)

Interactive Live Calculator & Visualizer

Chi-Square Test Engine (χ2\chi^2χ2)

Chi-Square (χ²) Test
Goodness of Fit Test

Chi-Square Test Engine ( $\chi^2$ )

Chi-Square (χ²) Test
Goodness of Fit Test

Chi-Square Test Engine ( $\chi^2$ )