ECON20003 · Quantitative Methods 2
Inferences on Variances & Proportions (χ², t, F)
Inferences on Variances & Proportions rounds out the single- and two-sample toolkit using the χ², t and F distributions. You test a single variance with a χ² statistic, compare two variances with the variance-ratio F-test (the gatekeeper for pooled vs Welch), and test one or two proportions with Z-statistics built on the pooled estimate. The chapter also covers χ² frequency analysis — goodness-of-fit and tests of independence in a contingency table — where you compare observed counts to expected counts.
What this chapter covers
- 01χ² distribution: E(V) = m, Var(V) = 2m; one-variance test χ² = (n−1)S²/σ₀²
- 02Confidence interval for σ² using χ² critical values
- 03Variance-ratio F-test: F = S₁²/S₂², df = (n₁−1, n₂−1)
- 04One-proportion Z-test: Z = (p̂ − p₀)/√(p₀q₀/n)
- 05Two-proportion Z-test with pooled p̂ = (f₁ + f₂)/(n₁ + n₂)
- 06χ² goodness-of-fit and test of independence; eᵢⱼ = (rowᵢ·colⱼ)/n
F-test for equal variances (the pooled-vs-Welch gatekeeper)
- 1 markState the hypotheses (two-tailed): H₀: σ₁² = σ₂² versus H₁: σ₁² ≠ σ₂².
- 2 marksForm the variance ratio with the LARGER variance on top so the statistic is ≥ 1: F = S₁²/S₂² = 48/20 = 2.4.
- 1 markDegrees of freedom: numerator df = n₁ − 1 = 15, denominator df = n₂ − 1 = 15.
- 1 markDecision rule: reject if F > F₀.₀₂₅,₁₅,₁₅ = 2.86 (two-tailed at 5%, so α/2 in the upper tail).
- 1 markCompare: 2.4 < 2.86, so do NOT reject H₀.
- 1 markConclude in context: there is insufficient evidence that the variances differ, so equal variances is reasonable — proceed with the pooled (equal-variance) t-test for any later mean comparison.
Key terms
- χ² (chi-square) distribution
- The distribution of a sum of squared standard normals; right-skewed with mean equal to its df and variance 2·df. It underlies tests of a single variance and frequency (goodness-of-fit and independence) tests.
- Variance-ratio F-test
- A test of H₀: σ₁² = σ₂² using F = S₁²/S₂² with df = (n₁−1, n₂−1). It is the gatekeeper deciding between the pooled and Welch two-sample t-tests.
- Pooled proportion
- When testing two proportions under H₀ of equality, combine both samples into p̂ = (f₁ + f₂)/(n₁ + n₂) to estimate the common proportion for the standard error of the Z-test.
- Expected frequency
- In a χ² contingency table, the count expected under independence, eᵢⱼ = (row total × column total)/grand total. The test sums (observed − expected)²/expected with df = (r − 1)(c − 1).
Inferences on Variances & Proportions (χ², t, F) FAQ
Why does R's two-proportion result not exactly match my hand Z²?
R reports a continuity-corrected χ² statistic by default, which nudges the value to account for using a continuous distribution to approximate discrete counts. Your uncorrected Z² will be close but not identical — both lead to the same decision in almost every case, so explain the discrepancy rather than treating it as an error.
How many degrees of freedom does a χ² frequency test have?
For goodness-of-fit, df = k − 1 − (number of parameters estimated from the data). For a test of independence in an r × c contingency table, df = (r − 1)(c − 1). Getting the df right is essential to reading the correct critical value.
Exam move
Keep the three sampling distributions — χ² (variances/frequencies), t (means), F (variance ratios and ANOVA) — and their df rules on a single index card. Practise the variance F-test as the hinge between the two-sample chapters, and rehearse building expected counts for a contingency table.
