26134 · Responsible Evidence-based Decisions
Testing Ideas II: Comparing Groups
Week 9 extends hypothesis testing to comparisons: the two-independent-samples t-test (with its equal-variance assumption), the paired-samples t-test as a one-sample test on differences, ANOVA for more than two groups, and the F-test for equal variances. The subject's decision tree routes each question to the right test — recognising paired vs independent, and choosing among t, F and ANOVA, is exactly what the exam checks.
What this chapter covers
- 01Two independent samples, difference in means: pooled variance s²_p and t with df = n₁ + n₂ − 2
- 02Equal-variance assumption for the pooled t-test
- 03Paired (dependent) samples: reduce to a one-sample t-test on the differences dᵢ, df = n − 1
- 04Recognising paired vs independent designs (before/after on the same units = paired)
- 05ANOVA for more than two means: SST = SSB + SSW, F = MSB/MSW
- 06F-test for equal variances: F = s₁²/s₂² (larger over smaller), df = (n₁ − 1, n₂ − 1)
- 07The decision tree for choosing a test
- 08The F-distribution: right-skewed, non-negative, two df parameters
Paired-samples t-test on before/after data
- +1Recognise the design and hypotheses. The two measurements are on the SAME people, so this is paired: reduce to a one-sample t-test on the differences. H₀: μ_d = 0 versus H₁: μ_d > 0 (a positive before − after difference means weight fell).
- +1Standard error of the mean difference. SE = s_d/√n = 3.0/√10 = 3.0/3.162 = 0.949 kg.
- +1Test statistic. t = d̄/SE = 2.4/0.949 = 2.53 on df = n − 1 = 9.
- +1Decision. One-tailed critical value t_(0.05, 9) = 1.833. Since 2.53 > 1.833, reject H₀: there is evidence at the 5% level that mean weight decreased over the programme.
Key terms
- Two independent-samples t-test
- Tests H₀: μ₁ = μ₂ for two separate groups. Under the taught equal-variance assumption it pools the two sample variances into s²_p and uses t = (x̄₁ − x̄₂)/√[s²_p(1/n₁ + 1/n₂)] with df = n₁ + n₂ − 2.
- Pooled variance
- s²_p = [(n₁ − 1)s₁² + (n₂ − 1)s₂²]/(n₁ + n₂ − 2), a weighted average of the two sample variances used when the two populations are assumed to share a common variance.
- Paired-samples t-test
- For naturally matched observations (before/after on the same units), it reduces to a one-sample t-test on the differences dᵢ: t = d̄/(s_d/√n), df = n − 1. Pairing removes between-subject variability.
- ANOVA (one-way)
- Analysis of Variance tests whether three or more group means are all equal by partitioning variability, SST = SSB (between) + SSW (within), and forming F = MSB/MSW. A large F is evidence that at least one mean differs.
- F-test for equal variances
- Tests H₀: σ₁² = σ₂² using the ratio F = s₁²/s₂² (conventionally the larger variance on top), with df = (n₁ − 1, n₂ − 1). A ratio far from 1 is evidence the variances differ.
- F-distribution
- A right-skewed, non-negative distribution with two degrees-of-freedom parameters (numerator and denominator). It arises for ratios of variances, so it underlies both ANOVA and the two-variance F-test.
Testing Ideas II: Comparing Groups FAQ
How do I tell a paired test from a two-independent-sample test?
Ask whether each observation in one group is naturally matched to a specific observation in the other. Before-and-after measurements on the same people, or matched pairs, are PAIRED — analyse the differences with a one-sample t-test. Two separate, unrelated groups (e.g. firm A vs firm B) are INDEPENDENT and use the pooled two-sample t-test.
When do I use ANOVA instead of a t-test?
Use a t-test to compare two means and ANOVA to compare three or more group means at once. Running many pairwise t-tests inflates the overall Type I error, whereas one ANOVA tests H₀ that all means are equal using the F-ratio MSB/MSW. A significant F says at least one mean differs, not which.
What is the F-test for, and why put the larger variance on top?
The F-test compares two variances (for example, which of two investments is more volatile), using F = s₁²/s₂². Placing the larger sample variance in the numerator keeps F ≥ 1, so you only need the upper tail of the F-table. A ratio far above 1 is evidence the variances differ.
Can AI help me with comparing groups in 26134?
Yes, as a study aid. Sia can walk you through the decision tree — paired vs independent, two groups vs many, means vs variances — and then through the chosen test's statistic and decision, step by step. Use it to rehearse the method; it does not do your graded work, and the UTS academic-integrity policy applies.
Exam move
Week 9 is really about routing, so master the decision tree first: one sample or two? Means, a variance, or several means? Paired or independent? Known or unknown σ? Then two groups (t or F) or more (ANOVA)? Practise classifying a scenario BEFORE computing, because choosing the wrong test is the biggest mark-loser. Drill the paired-vs-independent distinction and the pooled-variance formula, and know that ANOVA and the F-test both use the right-skewed F-distribution with two df. Keep the pooled t, paired t, ANOVA F and F-test formulas plus the decision tree on your printed exam notes. When you are unsure which test fits, ask Sia to route a fresh scenario with you and then execute it; confirm assessment details on Canvas.
Working through Testing Ideas II: Comparing Groups in 26134? Sia is AskSia’s AI Statistics tutor — ask any 26134 Testing Ideas II: Comparing Groups question and get a clear, step-by-step explanation grounded in how 26134 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
