Null Hypothesis Significance Testing – Quick Review

Hypothesis Testing Framework

  • Start with two competing statements about a population parameter (usually \mu)
  • Assume the null hypothesis is true; use sample data to challenge it
  • Goal: decide whether to reject H0 (supporting H1) or fail to reject H_0

Null vs. Alternative Hypotheses

  • Always created as a pair
    • H_0 (null/comparison): contains an equality (e.g., =, \le, \ge)
    • H_1 (alternative/research): contains the opposite inequality (e.g., \neq,
  • Must be mutually exclusive and collectively exhaustive
  • Example (non-directional):
    • H0: \mu{NYC} = 2.17
    • H1: \mu{NYC} \neq 2.17

Directionality

  • Non-Directional (Two-Tailed): test for any difference; H_1 uses \neq
  • Directional (One-Tailed): test for a specific direction; H_1 uses > or <
    • Example (greater-than): H0: \mu{Law} \le 21, H1: \mu{Law} > 21

Sampling Distribution & Standard Error

  • Under H_0, the sampling distribution of \bar x is normal
    • Mean = \mu_{0}
    • Standard error = \sigma/\sqrt{N}
  • Provides the reference to judge how extreme the sample mean is

Alpha (\alpha), p-Value, and Critical Values

  • Alpha (\alpha): researcher-set rejection threshold (commonly 0.05)
  • Critical value (CV): cutoff on the H_0 distribution that bounds the rejection region
    • One-tailed: single CV in the tail with area \alpha
    • Two-tailed: two symmetric CVs with total area \alpha (each tail \alpha/2)
  • p-value: probability of obtaining the observed statistic (or more extreme) given H_0 is true
    • Reject H_0 if p < \alpha (statistically significant)

Decision Outcomes

  • Reject H0: evidence favors H1; result is statistically significant
  • Fail to Reject H0: insufficient evidence against H0; result is not significant

Step-by-Step Summary

  1. State H0 and H1 (include directionality)
  2. Select \alpha
  3. Determine sampling distribution & critical value(s)
  4. Collect sample, compute statistic (e.g., \bar x)
  5. Compare statistic to CVs or compute p-value
  6. Make decision: reject or fail to reject H_0

Key Vocabulary

  • Null Hypothesis (H_0): status-quo/comparison claim
  • Alternative Hypothesis (H_1): researcher’s substantive claim
  • Alpha (\alpha): Type I error rate (probability of wrongly rejecting H_0)
  • p-value: observed extremity probability under H_0
  • Critical Value: boundary of the rejection region
  • Statistical Significance: when p < \alpha (or statistic falls in rejection region)