6.1 Formulation of null and alternative hypotheses

Hypothesis testing is all about making educated guesses about populations using sample data. We start by setting up two competing ideas: the null hypothesis (no effect) and the alternative hypothesis (there is an effect).

Formulating these hypotheses is crucial. We need to clearly state what we're testing, using the right statistical language. This sets the stage for the whole testing process, guiding how we'll collect and analyze our data.

Hypothesis testing components

Purpose and key elements

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• Hypothesis tests make inferences about population parameters using sample data
• Evaluate plausibility of specific claim (null hypothesis) about population parameter
• Test statistic calculated from sample data assesses evidence against null hypothesis
• Significance level (α) sets predetermined probability of incorrectly rejecting true null hypothesis (typically 0.05 or 0.01)
• P-value gives probability of obtaining test statistic as extreme or more extreme than observed, assuming null hypothesis is true
• Decision rule compares p-value to significance level or test statistic to critical values

Statistical inference process

• Formulate null and alternative hypotheses about population parameter
• Collect sample data and calculate relevant test statistic
• Determine distribution of test statistic under null hypothesis
• Define rejection region based on significance level
• Calculate p-value or compare test statistic to critical values
• Make decision to reject or fail to reject null hypothesis
• Interpret results in context of original research question

Null vs Alternative hypotheses

Defining characteristics

• Null hypothesis (H₀) states no effect, difference, or relationship between variables
• Alternative hypothesis (H₁ or Hₐ) contradicts null, suggesting effect exists
• Null always tested statement, alternative represents researcher's belief or hope
• Null typically uses equality (=, ≤, ≥), alternative uses inequality (≠, <, >)
• Hypotheses mutually exclusive and exhaustive, covering all possible outcomes
• Goal often to reject null in favor of alternative, providing evidence for significant effect

Hypothesis structure

• Two-tailed test null: equality (H₀: μ = μ₀), alternative: inequality (H₁: μ ≠ μ₀)
• One-tailed test null: equality and direction (H₀: μ ≤ μ₀ or H₀: μ ≥ μ₀), alternative: opposite direction (H₁: μ > μ₀ or H₁: μ < μ₀)
• Population comparison null: no difference (H₀: μ₁ = μ₂), alternative: difference exists (H₁: μ₁ ≠ μ₂)
• Correlation studies null: no correlation (H₀: ρ = 0), alternative: non-zero correlation (H₁: ρ ≠ 0)
• Categorical data null: independence between variables, alternative: association exists
• State hypotheses using population parameters (μ, σ, ρ) not sample statistics (x̄, s, r)

Formulating hypotheses for research

Research question translation

• Identify key variables and relationships in research question
• Determine appropriate population parameter to test (mean, proportion, correlation)
• Consider directionality of expected effect (increase, decrease, or any change)
• Choose between one-tailed and two-tailed tests based on research question and prior knowledge
• Ensure hypotheses address all aspects of research question
• Phrase hypotheses in clear, concise statistical language

Examples of hypothesis formulation

• Testing new drug effectiveness:
  • H₀: μ_treatment ≤ μ_placebo (new drug not more effective than placebo)
  • H₁: μ_treatment > μ_placebo (new drug more effective than placebo)
• Comparing exam scores between two teaching methods:
  • H₀: μ_method1 = μ_method2 (no difference in average scores)
  • H₁: μ_method1 ≠ μ_method2 (difference exists in average scores)
• Investigating relationship between study time and grades:
  • H₀: ρ = 0 (no correlation between study time and grades)
  • H₁: ρ ≠ 0 (correlation exists between study time and grades)
• Testing for gender bias in hiring:
  • H₀: p_male = p_female (equal hiring proportions for males and females)
  • H₁: p_male ≠ p_female (unequal hiring proportions for males and females)

Critical region and decision rule

Defining critical region

• Critical region contains test statistic values leading to null hypothesis rejection
• Critical value marks boundary between critical and non-critical regions
• Determined by significance level and test statistic's sampling distribution
• Two-tailed tests have two critical regions, one in each distribution tail
• One-tailed tests have single critical region in direction of alternative hypothesis
• Smaller significance level (α) results in smaller critical region and more conservative test

Applying decision rules

• Reject null hypothesis if test statistic falls within critical region
• Fail to reject null hypothesis if test statistic outside critical region
• Compare calculated p-value to predetermined significance level (α)
• Reject null if p-value < α, fail to reject if p-value ≥ α
• For z-tests, compare z-statistic to critical z-value (z_α or z_α/2)
• For t-tests, compare t-statistic to critical t-value based on degrees of freedom
• Interpret decision in context of original research question and practical significance