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 $H<em>0$ (supporting $H</em>1$) 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):
  • $H<em>0: \mu</em>{NYC} = 2.17$
  • $H<em>1: \mu</em>{NYC} \neq 2.17$

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): $H<em>0: \mu</em>{Law} \le 21$, $H<em>1: \mu</em>{Law} > 21$

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 $H<em>0$: evidence favors $H</em>1$; result is statistically significant
• Fail to Reject $H<em>0$: insufficient evidence against $H</em>0$; result is not significant

Step-by-Step Summary

1. State $H<em>0$ and $H</em>1$ (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)