UNIT 5 • TESTING OF HYPOTHESIS

F-Test
Variance Ratio Test

The F-Test compares the ratio of two sample variances $F = s_1^2 / s_2^2$ to test whether two independent populations have equal variability ( $\sigma_1^2 = \sigma_2^2$ )!

F-STATISTIC VARIANCE RATIO FORMULARIGHT-TAILED RATIO

F = \frac{s_1^2}{s_2^2} \quad \text{where } s_1^2 \ge s_2^2

1. Numerator (s₁²)Always place the larger sample variance in numerator (

s_1^2 \ge s_2^2

2. Degrees of FreedomNumerator

df_1 = n_1 - 1

, Denominator

df_2 = n_2 - 1

3. Critical DecisionReject

H_0

F_{calc} > F_{\alpha}(df_1, df_2)

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Compare variances of 2 independent sample groups:

n_1 = 10, \, s_1^2 = 25.0 \quad \text{vs} \quad n_2 = 12, \, s_2^2 = 10.0

Test the null hypothesis $H_0: \sigma_1^2 = \sigma_2^2$ at significance level $\alpha = 0.05$ !

STEP 01 • DEGREES OF FREEDOM & HYPOTHESES

Formulate hypotheses and calculate degrees of freedom:

H_0: \sigma_1^2 = \sigma_2^2, \quad H_1: \sigma_1^2 \neq \sigma_2^2

df_1 = n_1 - 1 = 9, \quad df_2 = n_2 - 1 = 11

STEP 02 • COMPUTE F-STATISTIC & CRITICAL THRESHOLD

Calculate variance ratio $F_{calc}$ :

F_{\text{calc}} = \frac{s_1^2}{s_2^2} = \frac{25.0}{10.0} = 2.50

F_{\text{calc}} = 2.50 < F_{0.05}(9, 11) = 2.896

FINAL RESULT

Decision: Fail to Reject H₀ (Variances Not Significantly Different)

\boxed{\text{Fail to Reject } H_0: F_{\text{calc}} = 2.50 < 2.896}

Because calculated $F_{calc} = 2.50$ is less than critical value $2.896$ , we fail to reject the null hypothesis—concluding there is no significant difference between population variances!

LABORATORY

Interactive Live Calculator & Visualizer

VARIANCE RATIO LABORATORY

Snedecor's F-Test Engine (Two-Sample Variance)

PreviousChi-Square Test

Unit 5•28 of 28

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 5 • TESTING OF HYPOTHESIS

F-Test
Variance Ratio Test

The F-Test compares the ratio of two sample variances $F = s_1^2 / s_2^2$ to test whether two independent populations have equal variability ( $\sigma_1^2 = \sigma_2^2$ )!

F-STATISTIC VARIANCE RATIO FORMULARIGHT-TAILED RATIO

F = \frac{s_1^2}{s_2^2} \quad \text{where } s_1^2 \ge s_2^2

1. Numerator (s₁²)Always place the larger sample variance in numerator (

s_1^2 \ge s_2^2

2. Degrees of FreedomNumerator

df_1 = n_1 - 1

, Denominator

df_2 = n_2 - 1

3. Critical DecisionReject

H_0

F_{calc} > F_{\alpha}(df_1, df_2)

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Compare variances of 2 independent sample groups:

n_1 = 10, \, s_1^2 = 25.0 \quad \text{vs} \quad n_2 = 12, \, s_2^2 = 10.0

Test the null hypothesis $H_0: \sigma_1^2 = \sigma_2^2$ at significance level $\alpha = 0.05$ !

STEP 01 • DEGREES OF FREEDOM & HYPOTHESES

Formulate hypotheses and calculate degrees of freedom:

H_0: \sigma_1^2 = \sigma_2^2, \quad H_1: \sigma_1^2 \neq \sigma_2^2

df_1 = n_1 - 1 = 9, \quad df_2 = n_2 - 1 = 11

STEP 02 • COMPUTE F-STATISTIC & CRITICAL THRESHOLD

Calculate variance ratio $F_{calc}$ :

F_{\text{calc}} = \frac{s_1^2}{s_2^2} = \frac{25.0}{10.0} = 2.50

F_{\text{calc}} = 2.50 < F_{0.05}(9, 11) = 2.896

FINAL RESULT

Decision: Fail to Reject H₀ (Variances Not Significantly Different)

\boxed{\text{Fail to Reject } H_0: F_{\text{calc}} = 2.50 < 2.896}

Because calculated $F_{calc} = 2.50$ is less than critical value $2.896$ , we fail to reject the null hypothesis—concluding there is no significant difference between population variances!

LABORATORY

Interactive Live Calculator & Visualizer

VARIANCE RATIO LABORATORY

Snedecor's F-Test Engine (Two-Sample Variance)

PreviousChi-Square Test

Unit 5•28 of 28

F-Test Variance Ratio Test

Step-by-Step Worked Example

Decision: Fail to Reject H₀ (Variances Not Significantly Different)

Interactive Live Calculator & Visualizer

Snedecor's F-Test Engine (Two-Sample Variance)

NETZ Engine

F-Test Variance Ratio Test

Step-by-Step Worked Example

Decision: Fail to Reject H₀ (Variances Not Significantly Different)

Interactive Live Calculator & Visualizer

Snedecor's F-Test Engine (Two-Sample Variance)

F-Test
Variance Ratio Test

F-Test
Variance Ratio Test