Week 4 Lecture Notes - Introduction to t-tests

Week 4 Lecture 1: Introduction to t-tests
Overview of Ethics in Research
  • Significance of Ethics Committees
    • Existence traced back to historical documents:
    • Nuremberg Code (1947)
    • Universal Declaration of Human Rights (1948)
    • Belmont Report (1979)
    • Declaration of Helsinki (1964)
Aotearoa New Zealand Research Ethics
  • Research must align with:
    • The Treaty of Waitangi
    • Māori Data Sovereignty Principles
  • False statements about New Zealand:
    • Registered psychologists must adhere to ethical principles laid out by the New Zealand Psychologists Board
    • The New Zealand Human Rights Commission oversees international human rights commitments
t-tests Overview
  • Types of Hypotheses:
    • Null Hypothesis (H₀) vs Alternative Hypothesis (H₁)
    • Directional vs Non-directional
  • Hypothesis Testing Workflow:
    1. State the Hypothesis
    2. Determine Significance Level
    3. Compute Test Statistics
    4. Calculate P-value
    5. Compare P-value with Significance Level
    6. Make a Decision to Reject or Fail to Reject H₀
Directional Hypothesis Testing
  • A directional hypothesis predicts the specific direction of the effect (e.g., Mean > 0)
  • Corresponding Null Hypothesis (e.g., Mean ≤ 0)
  • One-sided test: Focuses on one direction of the critical region (e.g., top 5% of distribution)
Non-Directional Hypothesis Testing
  • Non-directional hypotheses do not predict a specific direction (e.g., Mean ≠ 0)
  • Corresponding Null Hypothesis: Mean = 0
  • Two-sided test: Looks in both tails of distribution (upper and lower 2.5%)
One-Sample t-Test
  • Used to compare the sample mean against a population mean
  • Assumptions:
    1. Random sample
    2. Normal distribution of data (can use Shapiro-Wilk test)
Example Research Question
  • Do children in multilingual homes have larger or smaller vocabularies than average?
  • Prior research suggests average child knows 300 words; sample from multilingual homes taken
    • Null Hypothesis (H₀): Mean = 300
    • Alternative Hypothesis (H₁): Mean ≠ 300
Calculating t-statistic

  • Formula for t-statistic:

    t-statistic formula

  • Based on:

    • Sample mean
    • Population mean under H₀
    • Sample standard deviation
    • Sample size
Effect Size Calculation

  • Cohen's d:

    Cohen's d

    • Indicates the magnitude of the difference between groups
    • Important only when there's significant evidence of an effect
Reporting Results
  • Example statement:
    • "The mean vocabulary size in multilingual two-year-olds was 289.28 words (SD = 49.91). A two-tailed one-sample t-test found no evidence that this differs from the average vocabulary size in the wider population of 300 words (t(59) = -1.69, p = .096, n.s., α = 0.05)."
Non-Parametric Tests as Alternatives
  • Use when assumptions of t-tests not met
  • Wilcoxon Test:
    • Compares central tendency without assuming normality
    • Follows similar reporting structure as t-test but does not require degrees of freedom
  • Example statement:
    • "The mean vocabulary size in multilingual two-year-olds was 289.28 words (SD = 49.91) … (V = 8.5, p = 0.397, n.s., α = 0.05)."