Section 11.4

Inference about Two Population Standard Deviations

Compare variability between two populations using the F-distribution.

1

The F-Distribution

Values are always ≥ 0

Not symmetric (right-skewed)

Two df values: (numerator) and (denominator)

2

Requirements

Extremely Sensitive: This test is not robust to departures from normality. Both populations must be normal.

Independent random samples
Both populations normally distributed
3

F-Test Statistic

Convention: Place the larger sample variance in the numerator so

4

F-Test Calculator

F-Test Calculator (Two Variances)

Compare variability between two populations

Warning: This test is extremely sensitive to departures from normality. Both populations must be normally distributed.

Sample 1

Sample 2

Null:
Alternative:
df₁:
19
df₂:
24

Calculations

s₁²
156.2500
s₂²
68.8900
df₁
19
df₂
24
Test Statistic

F-Distribution (df₁ = 19, df₂ = 24)

0.4082.3451F₀ = 2.27

Classical Approach

Critical Region: F < 0.4078 or F > 2.3452
F₀ = 2.2681 is NOT in rejection region

P-Value Approach

P-value = 0.0595
P-value ≥ α = 0.05

Fail to Reject H₀

At the α = 0.05 level, there is insufficient evidence to conclude that the two population variances are different.

!

Common Pitfalls

Ignoring Normality Requirement

The F-test is extremely sensitive to non-normality. Verify normality first.

Using Standard Deviations

The F-test uses variances (), not standard deviations.

Wrong df Order

df₁ (numerator) and df₂ (denominator) are not interchangeable.

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