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Hypothesis Writing and Testing

Aug 4, 2025

Overview

This lecture covers how to identify and write null and alternative hypotheses, determine their roles in statistical testing, and select the correct type of hypothesis test.

Null and Alternative Hypotheses

  • The null hypothesis (H₀) always includes the equal sign (e.g., μ = value).
  • The alternative hypothesis (H₁ or Hₐ) uses symbols like ≠, <, or > and never contains the equal sign.
  • Write the claim, then the opposing view, and assign H₀ and Hₐ based on the presence of the equal sign.
  • The claim itself does not determine which is null or alternative; focus on the mathematical sign.

Identifying and Writing Hypotheses

  • If the claim is "μ < 98.6", the opposing view is "μ = 98.6"; H₀: μ = 98.6, Hₐ: μ < 98.6.
  • If the claim is "μ = 100", the opposing view is "μ ≠ 100"; H₀: μ = 100, Hₐ: μ ≠ 100.
  • Mathematical opposites aren't required; the null always has the equal sign, alternatives do not.

Determining Test Types

  • The form of the alternative hypothesis (Hₐ) determines the test type:
    • Hₐ: < (less than) leads to a left-tailed test.
    • Hₐ: > (greater than) leads to a right-tailed test.
    • Hₐ: ≠ (not equal) leads to a two-tailed test.
  • The test direction affects how you interpret the p-value and test statistics.

Writing Hypotheses for Multiple Populations

  • For two population tests, place population 1 on the left and population 2 on the right (e.g., μ₁ < μ₂ for group comparisons).
  • Label groups clearly according to the context provided (e.g., men as group 1, women as group 2).

Steps for Hypothesis Testing

  • Step 1: Write down the claim.
  • Step 2: Write the opposing view.
  • Step 3: Assign which is H₀ and which is Hₐ based on the presence of the equal sign.

Key Terms & Definitions

  • Null Hypothesis (H₀) — A statement with an equal sign, representing no change or default status.
  • Alternative Hypothesis (Hₐ) — A statement using ≠, <, or >, representing a suspected difference or effect.
  • Left-tailed Test — A hypothesis test where Hₐ uses "<".
  • Right-tailed Test — A hypothesis test where Hₐ uses ">".
  • Two-tailed Test — A hypothesis test where Hₐ uses "≠".

Action Items / Next Steps

  • Practice writing null and alternative hypotheses for various claims.
  • Label group comparisons properly in two population problems.
  • Review test types decided by the alternative hypothesis for upcoming exercises.

Full transcript