Types of Two-Sample T-Tests

Jul 29, 2025

Overview

This lecture covers the three types of two-sample t-tests: pooled t-test (equal variances), t-test with unequal variances, and paired t-test (dependent samples), including their assumptions, formulas, step-by-step procedures, and example problems.

Types of Two-Sample t-Tests

  • Pooled t-test: Used when comparing means from two independent samples assuming unknown but equal population standard deviations.
  • Unequal variance t-test: Used when population standard deviations are unknown and assumed unequal.
  • Paired t-test: Used for dependent samples, typically the same subjects measured twice or matched pairs.

Pooled t-Test (Equal Variances)

  • Assumptions: Normal populations, equal but unknown standard deviations, and independent samples.
  • Pooled variance (Sp²) formula:
    Sp² = [(n1-1)S1² + (n2-1)S2²] / (n1 + n2 - 2)
  • t-statistic formula:
    t = (X̄1 - X̄2) / sqrt[Sp²(1/n1 + 1/n2)]
  • Degrees of freedom = n1 + n2 - 2
  • Hypotheses typically set as H0: μ1 = μ2, H1: μ1 ≠ μ2 (two-tailed when no direction is specified)
  • Use t-table to compare calculated t to critical values; fail to reject H0 if t falls between critical values.

Unequal Variance t-Test

  • Used when standard deviations are unknown and unequal.
  • t-statistic formula:
    t = (X̄1 - X̄2) / sqrt[(S1²/n1) + (S2²/n2)]
  • Degrees of freedom calculated using a specific formula (rounded down).
  • Same hypothesis setup and comparison to critical values as pooled test.

Paired t-Test (Dependent Samples)

  • Used for repeated measures on same subjects or matched pairs.
  • t-statistic formula:
    t = DÌ„ / (Sd / sqrt(n)), where DÌ„ is mean difference and Sd is standard deviation of the differences.
  • Degrees of freedom = n - 1 (n = number of pairs).
  • Hypotheses: H0: DÌ„ = 0, H1: DÌ„ ≠ 0.
  • Reduces variation in sampling distribution compared to independent samples.

Procedure for All t-Tests

  • Step 1: State null and alternative hypotheses.
  • Step 2: Set significance level (α).
  • Step 3: Identify appropriate t-test type.
  • Step 4: Determine decision rule using t-table (critical value).
  • Step 5: Calculate test statistic.
  • Step 6: Find p-value and compare with α to make decision.

Key Terms & Definitions

  • Pooled variance (Sp²) — Weighted average of sample variances under equal variance assumption.
  • Degrees of freedom — Number of independent observations minus the number of estimated parameters.
  • t-statistic — Test statistic used to compare means in t-tests.
  • p-value — Probability of obtaining results at least as extreme as those observed, under H0.
  • Dependent samples — Samples where observations are paired or matched in some way.

Action Items / Next Steps

  • Complete the practice and connect problems assigned for this chapter.
  • Review the five two-sample test summary document uploaded by your instructor.
  • Prepare for additional practice problems in the next Zoom session.

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