Section 10.4

Hypothesis Tests for a Population Standard Deviation

Use the chi-square distribution to test claims about population variability.

1

Model Requirements

Critical: This test is NOT robust to departures from normality. The data must come from a normally distributed population.

Random sample
Population is normally distributed (required!)
2

Chi-Square Test Statistic

Degrees of Freedom:

3

Rejection Regions

The chi-square distribution is not symmetric. Critical values depend on the type of test:

Two-tailed ()

Reject if or

Left-tailed ()

Reject if

Right-tailed ()

Reject if

4

χ²-Test Calculator

Hypothesis Test Calculator for Population Standard Deviation (χ²-test)

Test claims about population variability using the chi-square distribution

Important: This test requires the population to be normally distributed. It is NOT robust to departures from normality.

Enter:

Hypotheses

Null:
Alternative:
df:
24

Calculations

Sample Variance
Sample Std Dev
Hypothesized Variance
Test Statistic

χ² Distribution (df = 24)

12.4139.35χ² = 34.56RejectRejectFail to Reject

Classical Approach

Critical Region: χ² < 12.414 or χ² > 39.351
χ₀² = 34.5600 is NOT in rejection region

P-Value Approach

P-value = 0.1504
P-value ≥ α = 0.05

Fail to Reject H₀

At the α = 0.05 significance level, there is insufficient evidence to conclude that the population standard deviation differs from 2.

!

Common Pitfalls

Assuming Robustness

Unlike the t-test, the chi-square test for σ is very sensitive to non-normality.

Using Instead of

The formula uses variances ( and ), not standard deviations.

Treating χ² as Symmetric

The chi-square distribution is right-skewed. Two-tailed critical values are different.

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