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Chapter 9: the t Statistic - Coggle Diagram
Chapter 9: the t Statistic
Purpose & Evolution
Used when population variance (σ²) or standard deviation (σ) is unknown
Replaces the z-score test when only sample data is available
The t-Formula
Formula: t = (M - μ) / s_M
Sample Mean (M): Obtained directly from active sample data
Population Mean (μ): Hypothesized value from the null hypothesis
Estimated Standard Error (s_M): Expected distance between M and μ
Standard Error & Variance
Error Formula 1: s_M = s / √n
Error Formula 2: s_M = √(s² / n)
Sample Variance (s²): s² = SS / (n - 1)
Standard Deviation (s): s = √s²
Degrees of Freedom (df)
Formula: df = n - 1
Definition: Number of scores in a sample free to vary
Role: Directly determines the shape of the t-distribution
Constraint: Restricts the sample variability estimation
Distribution Comparison
Similarities: Symmetrical, bell-shaped, and centered at a mean of 0
Differences: t-distributions have flatter profiles and thicker tails
Sample Size Effect: Larger sample sizes (higher df) make the t-shape match the z-distribution
Variability: Has more error than z because variance is estimated, not known
Hypothesis Testing
Step 1: State the null (H0) and alternative (H1) hypotheses
Step 2: Locate the critical boundaries using alpha and df in the t-table
Step 3: Compute the sample variance, standard error, and t-statistic
Step 4: Make a decision to reject or fail to reject H0
Effect Size
Estimated Cohen's d: d = (M - μ) / s
Magnitude: Measures size of treatment effect without sample size bias
Percentage of Variance (r²): r² = t² / (t² + df)
Interpretation: Proportions of total variability explained by the treatment