Statistics

8.

Inferential Statistics: Hypothesis Testing

8.1.

The Logic of Hypothesis Testing

8.1.1.

Scientific Method Connection

8.1.2.

Proof by Contradiction

8.1.3.

Null Hypothesis (H₀)

8.1.3.1.

8.1.3.2.

No Effect Hypothesis

8.1.4.

Alternative Hypothesis (H₁)

8.1.4.1.

Research Hypothesis

8.1.4.2.

What We Want to Prove

8.1.5.

One-Tailed vs. Two-Tailed Tests

8.1.5.1.

Directional Hypotheses

8.1.5.2.

Non-Directional Hypotheses

8.1.5.3.

Critical Region Placement

8.2.

Errors in Hypothesis Testing

8.2.1.

Type I Error (α)

8.2.1.1.

8.2.1.2.

8.2.1.3.

Significance Level

8.2.2.

Type II Error (β)

8.2.2.1.

8.2.2.2.

8.2.2.3.

Factors Affecting Beta

8.2.3.

Power of a Test (1 - β)

8.2.3.1.

Factors Affecting Power

8.2.3.1.1.

8.2.3.1.2.

8.2.3.1.3.

Significance Level

8.2.3.1.4.

Population Variability

8.2.4.

Relationship Between Errors

8.2.4.1.

Trade-off Between Type I and Type II

8.3.

The Hypothesis Testing Process

8.3.1.

Stating Hypotheses

8.3.1.1.

Null Hypothesis Formulation

8.3.1.2.

Alternative Hypothesis Formulation

8.3.2.

Selecting a Significance Level (α)

8.3.2.1.

8.3.2.2.

Practical Considerations

8.3.3.

Identifying the Test Statistic

8.3.3.1.

Appropriate Test Selection

8.3.3.2.

Distribution Assumptions

8.3.4.

Formulating a Decision Rule

8.3.4.1.

Critical Value Method

8.3.4.2.

Rejection Region

8.3.5.

Calculating the Test Statistic

8.3.5.1.

Formula Application

8.3.5.2.

Computational Steps

8.3.6.

Making a Statistical Decision

8.3.6.1.

Critical Value Approach

8.3.6.1.1.

Comparison with Critical Values

8.3.6.1.2.

8.3.6.2.

P-Value Approach

8.3.6.2.1.

P-Value Calculation

8.3.6.2.2.

8.3.6.2.3.

8.3.7.

Interpreting Results

8.3.7.1.

Statistical vs. Practical Significance

8.3.7.2.

Context Considerations

8.4.

Hypothesis Tests for a Single Population

8.4.1.

Test for a Population Mean

8.4.1.1.

Z-test (σ known)

8.4.1.1.1.

8.4.1.1.2.

8.4.1.1.3.

8.4.1.2.

t-test (σ unknown)

8.4.1.2.1.

8.4.1.2.2.

8.4.1.2.3.

Degrees of Freedom

8.4.1.2.4.

8.4.2.

Test for a Population Proportion (Z-test)

8.4.2.1.

8.4.2.2.

8.4.2.3.

Sample Size Requirements

8.4.2.4.

8.4.3.

Test for a Population Variance (Chi-Square test)

8.4.3.1.

8.4.3.2.

8.4.3.3.

Degrees of Freedom

8.4.3.4.

8.5.

Hypothesis Tests for Two Populations

8.5.1.

Comparing Two Independent Population Means

8.5.1.1.

Equal Variances Assumed

8.5.1.1.1.

8.5.1.1.2.

8.5.1.2.

Unequal Variances

8.5.1.2.1.

8.5.1.2.2.

Degrees of Freedom Adjustment

8.5.1.3.

Large Sample Z-test

8.5.2.

Comparing Two Dependent Population Means (Paired t-test)

8.5.2.1.

8.5.2.2.

Difference Scores

8.5.2.3.

8.5.2.4.

8.5.3.

Comparing Two Population Proportions

8.5.3.1.

Independent Samples

8.5.3.2.

8.5.3.3.

Pooled Proportion

8.5.3.4.

8.5.3.5.

8.5.4.

Comparing Two Population Variances (F-test)

8.5.4.1.

8.5.4.2.

8.5.4.3.

Degrees of Freedom

8.5.4.4.

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9. Analysis of Variance (ANOVA)