Null and alternative hypotheses form the foundation of statistical inference. They help researchers frame their questions and guide the analysis process. Understanding these concepts is crucial for interpreting study results and drawing meaningful conclusions.

Formulating hypotheses involves identifying research questions, determining parameters of interest, and expressing them mathematically. One-tailed and two-tailed tests, along with test statistics, further refine the analysis approach and quantify evidence against the null hypothesis.

Understanding Null and Alternative Hypotheses

Null and alternative hypotheses

• Null hypothesis (H₀) asserts no effect or difference, assumed true until evidence suggests otherwise represents status quo (Earth is flat)
• Alternative hypothesis (H₁ or Hₐ) posits effect or difference, challenges null hypothesis represents new theory or research claim (Earth is round)
• Framework for statistical decision-making guides interpretation of results and conclusions
• Basis for calculating probabilities and p-values in hypothesis testing

Formulation of hypotheses

1. Identify research question or claim (Does new drug lower blood pressure?)
2. Determine parameter of interest (mean blood pressure reduction)
3. Express null hypothesis as equality (H₀: μ = 0 mm Hg)
4. State alternative hypothesis as inequality (H₁: μ < 0 mm Hg)

• Population mean: H₀: μ = μ₀ vs. H₁: μ ≠ μ₀ (average height of adults)
• Population proportion: H₀: p ≤ p₀ vs. H₁: p > p₀ (percentage of voters supporting a candidate)
• Difference between means: H₀: μ₁ - μ₂ = 0 vs. H₁: μ₁ - μ₂ ≠ 0 (comparing effectiveness of two treatments)

One-tailed vs two-tailed hypotheses

• Two-tailed alternative hypothesis tests for difference in both directions uses ≠ symbol (new teaching method affects test scores)
• One-tailed alternative hypothesis tests for difference in one specific direction uses > or < symbols (new diet decreases weight)
• Choice between one-tailed and two-tailed tests depends on research question, prior knowledge, consequences of errors, and statistical power

Test statistics in hypothesis testing

• Test statistic quantifies difference between observed data and null hypothesis (z-score, t-statistic, chi-square)
• Calculation based on sample data and null hypothesis assumptions standardized to known probability distribution
• Measures evidence against null hypothesis large absolute values suggest stronger evidence for alternative
• Critical regions determined by significance level (α) and type of alternative hypothesis correspond to rejection of null hypothesis