Tests for Means, Variances & Proportions

Tests for Means, Variances & Proportions

Quick Answer: Choose the hypothesis test from the question and data type: t-tests for continuous means (including paired differences), F-test for two variances, ANOVA for means across three or more groups, and chi-square for counts/proportions in categories—then interpret p-values with effect size and process meaning, not letters alone.

Selecting the Right Test

Verify independence, measurement quality, and rough distribution assumptions. Software prints p-values; your job is correct setup and interpretation.

Tests for Means — t Family

One-sample t-test
H₀: μ = μ₀ vs Hₐ: μ ≠ μ₀ (or one-sided). Uses sample mean, s, and n. Compare a process mean to a historical standard or mid-target as a mean test (capability still needs the full distribution vs specs).

Two-sample (independent) t-test
H₀: μ₁ = μ₂ for two independent groups (Line A vs B, Supplier 1 vs 2). If variances differ substantially, avoid forcing a pooled variance (e.g., use a Welch-type procedure). Check that “group” is not confounded with time or operator.

Paired-comparison t-test
When each observation in one sample is naturally linked to one in the other, analyze the differences dᵢ:

H₀: μ_d = 0 (mean difference is zero)

Classic pairing: same part with Method 1 and Method 2; same agent before/after training; left/right on the same board. Pairing removes unit-to-unit noise, raising power for a true method effect.

Worked scenario — paired t
Ten machines timed on old and new setups. Differences (new − old): d̄ = −1.8 min, s_d = 1.5 min.

• SE ≈ 1.5 / √10 ≈ 0.47
• t ≈ −1.8 / 0.47 ≈ −3.8 (df = 9)

At α = 0.05 two-sided, |t| this large yields p well below 0.05 → reject H₀ of zero mean difference. Interpretation: average cycle time is lower on the new setup. Still judge practical significance (−1.8 min vs labor cost) and confirm stability after rollout.

Treating the same data as independent two-sample (ignoring pairing) inflates error with between-machine noise and can hide the setup effect—a common exam trap.

F-Test for Variances

F-test (two variances) compares s₁² and s₂²:

H₀: σ₁² = σ₂² vs Hₐ: not equal (or one-sided)

Statistic form: F = s₁² / s₂² (df = n₁−1, n₂−1), using the software’s ratio convention.

When it matters: choosing pooled vs unpooled t procedures; confirming a noise-reduction project cut variance; checking multi-vari hunches that one stream is wider.

Worked scenario — F-test
Shift A: n = 20, s = 2.0; Shift B: n = 18, s = 3.5.
F ≈ (3.5)² / (2.0)² = 3.06. For typical α and df, a ratio this large often gives p < 0.05 → reject equal variances. Interpretation: Shift B is more variable—investigate practices, staffing, or inputs, not only mean location.

Classical F-tests are sensitive to non-normality. Use histograms/boxplots; consider robust alternatives when outliers dominate.

ANOVA — Means of Three or More Groups

One-way ANOVA tests:

H₀: μ₁ = μ₂ = … = μₖ
Hₐ: at least one mean differs

It partitions variation into between-group and within-group mean squares; F = MS_between / MS_within. A significant F does not say which pairs differ—only that all-means-equal is untenable. Follow with plots, process knowledge, or planned comparisons.

Worked scenario — ANOVA
Four packaging lines’ seal strengths compared. ANOVA p = 0.008 at α = 0.05 → reject H₀. Boxplots show Line 3 below the others. Next: confirm Line 3 settings/maintenance—not automatically retool every line. If p = 0.40, fail to reject equal means—still check variance and practical gaps.

ANOVA assumes independent errors, similar within-group variances, and approximate residual normality; large equal samples are more robust.

Chi-Square — Proportions and Categories

Use counts, not continuous measurements.

1. Goodness-of-fit: Do observed frequencies match a stated distribution?
2. Independence / association (contingency table): Is defect type independent of shift? Pass/fail independent of supplier?

Statistic: χ² = Σ (O − E)² / E. Small expected counts weaken the approximation—combine categories or collect more data.

Worked scenario — chi-square
Scratch / Dent / Other by Day vs Night forms a 3×2 table; expected counts ≥ 5. χ² p = 0.03 → reject independence at α = 0.05. Interpretation: defect mix differs by shift—investigate lighting, handling, or inspection standards. This is not a t-test on percent defective unless you restructure properly (e.g., two-proportion methods with adequate n).

For simple two-proportion questions, CSSGB items may use a two-proportion z-test or χ² on a 2×2 table—same idea: small p → evidence proportions differ.

Putting Interpretation Together

Pitfalls and Exam Focus

• Using independent two-sample t when pairing is the design.
• Running many unplanned t-tests among k groups instead of ANOVA first.
• Applying chi-square to averages without raw counts.
• Reading a significant F or ANOVA as root cause without process investigation.
• Ignoring assumptions when n is tiny or outliers dominate.

CSSGB IV.B.2 items usually present a short scenario and ask which test, what H₀ means, or what a significant result allows—ground answers in the process question, not the option letter.