Day 22

Math 216: Statistical Thinking

Bastola

Introduction to Hypothesis Testing: Statistical Decision Framework

Key Question: How can we make objective decisions about population parameters using sample evidence? Hypothesis testing provides the mathematical framework for evaluating claims with quantified uncertainty!

Real-World Applications:

  • Quality Control: Verify manufacturing standards compliance
  • Medical Research: Test treatment effectiveness in clinical trials
  • Market Research: Evaluate consumer preference claims
  • Environmental Science: Assess regulatory compliance
  • Public Policy: Evaluate program effectiveness

Elements of a Test of Hypothesis

Real-World Case Study: Quality Control

Context: City regulations require residential sewer pipes to have an average breaking strength greater than 2,400 pounds per foot. Manufacturers must demonstrate that their products meet this standard.

Statistical Framework:

  • Null Hypothesis (\(H_0\)): \(\mu \leq 2400\) (Pipes do not meet standard)
    • Interpretation: Assume pipes are inadequate until proven otherwise
    • Analogy: “Innocent until proven guilty”
  • Alternative Hypothesis (\(H_a\)): \(\mu > 2400\) (Pipes meet or exceed standard)
    • Interpretation: What we want to demonstrate
    • Burden of Proof: Evidence must strongly support this claim

Key Insight: The null hypothesis represents the “status quo” or default position that requires strong evidence to overturn!

Testing the Hypothesis

Test Statistic Calculation

Central Limit Theorem Foundation: For large samples (\(n \geq 30\)), \(\bar{x}\) approximates a normal distribution, enabling standardized test statistics.

Test Statistic Formula: \[ z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} \]

Components:

  • \(\bar{x}\): Sample mean from data
  • \(\mu_0\): Hypothesized population mean (2400)
  • \(\sigma\): Population standard deviation
  • \(n\): Sample size

Interpretation: The z-score measures how many standard errors the sample mean is from the hypothesized value!

Decision Making Framework

Statistical Decision Criteria

Critical Value Approach:

  • If the observed \(\bar{x}\) exceeds 1.645 standard deviations above 2,400, we consider rejecting \(H_0\)
  • This corresponds to a 5% significance level (\(\alpha = 0.05\))

Statistical Reasoning:

  • Rejection implies that a rare event (occurring with probability \(\leq 0.05\) under \(H_0\)) is unlikely without true cause
  • If such an extreme result occurs, it provides evidence that \(H_a\) is likely true

Key Principle: We only reject \(H_0\) when the evidence is sufficiently strong to overcome the presumption of innocence!

Evidence from Sample Data

Practical Application: Pipe Strength Analysis

Sample Characteristics:

  • Number of samples (\(n\)): 50
  • Mean strength (\(\bar{x}\)): 2,460 pounds per linear foot
  • Standard deviation (\(s\)): 200 pounds per linear foot

Test Statistic Calculation: \[ z = \frac{\bar{x} - 2400}{s/\sqrt{n}} \approx \frac{2460 - 2400}{200 / \sqrt{50}} = \frac{60}{28.28} \approx 2.12 \]

Statistical Interpretation:

  • The sample mean is 2.12 standard errors above the hypothesized value
  • This exceeds the critical value of 1.645 for \(\alpha = 0.05\)
  • Conclusion: Strong evidence to reject \(H_0\) and conclude pipes meet the standard

Types of Errors

Type I Error

  • Analogy: Convicting an innocent person.
  • Occurs if we reject \(H_0\) when it is, in fact, true.
  • Probability of Type I error is denoted by \(\alpha\) (commonly set at 0.05).

Types of Errors

Type II Error

  • Analogy: Acquitting a guilty person.
  • Occurs if we fail to reject \(H_0\) when, in fact, \(H_a\) is true.
  • Probability of Type II error is denoted by \(\beta\).
  • Unlike \(\alpha\), \(\beta\) is not usually pre-specified but affected by the sample size, effect size, and the set \(\alpha\).

Types of Errors


Key Statistical Principles

Essential Hypothesis Testing Concepts:

  1. Presumption of Innocence: \(H_0\) represents the default position requiring strong evidence to overturn
  2. Burden of Proof: Evidence must be sufficiently strong to reject \(H_0\)
  3. Error Control: Balance Type I (\(\alpha\)) and Type II (\(\beta\)) errors
  4. Decision Framework: Use standardized test statistics and critical values

Statistical Guidelines:

  • Significance Level: Choose \(\alpha\) appropriate for research context
  • Test Statistic: Calculate standardized measure of evidence
  • Decision Rule: Compare test statistic to critical value
  • Interpretation: Focus on strength of evidence, not absolute certainty

Next Topic: Extending these principles to different types of hypothesis tests