Hypothesis Testing

Foundations of Hypothesis Testing

  • Hypothesis Testing Overview: Involves making a claim about a population parameter (e.g., mean), followed by statistical analysis to test this claim.

  • Types of Hypotheses:

    • Null Hypothesis (H0): Assumes no effect or difference; proposes a specific parameter value.

    • Alternate Hypothesis (H1): Contradicts the null hypothesis; indicates the presence of an effect or difference.

  • Testing Types:

    • One-Tail Test: Tests for effects in one direction.

    • Two-Tail Test: Tests for effects in both directions.

  • Errors in Hypothesis Testing:

    • Type I Error (⍺ Error): Rejecting H0 when it is true.

    • Type II Error (β Error): Failing to reject H0 when it is false.

  • Decision Rules:

    • Define a level of significance (common values: ⍺ = 0.05).

    • Determine the test statistic and critical value to compare with.

  • Test Statistics:

    • Z-test: Used when population variance ($\sigma$) is known and sample size (n) is large.

    • T-test: Used when $\sigma$ is unknown, n is small, and sample is from a normally distributed population.

  • Degrees of Freedom (df): Given by $df = n - 1$; one df is lost for each sample statistic used as a point estimator.

  • Using t-Distribution:

    • More spread than normal distribution, symmetrical about the mean; approaches normal as n increases.

  • Trade-offs in Errors: Increasing confidence (lowering ⍺) raises the risk of Type II errors (β).

  • Steps in Hypothesis Testing:

    1. Formulate H0 and H1.

    2. Specify significance level (⍺).

    3. Select test statistic (Z, t, etc.), establish critical values.

    4. Compute test statistic and make a decision - reject or do not reject H0.

    5. Take action based on results (business decision).