Comprehensive Study Guide: Hypothesis Testing
Complete reference for Module 8 concepts and formulas
1. Introduction to Hypothesis Testing
What is Hypothesis Testing?
A statistical procedure used to test a claim about a population parameter based on sample data. It uses proof by contradiction: assume a claim is true, collect data, and determine how likely the data would be if the claim were true.
Null and Alternative Hypotheses
| Hypothesis | Description | Symbol |
|---|---|---|
| Null (H₀) | The claim being tested; represents "no effect" or status quo. Always contains equals sign (=). | H₀: μ = μ₀ H₀: p = p₀ |
| Alternative (Hₐ or H₁) | The claim we're trying to find evidence for. Contains ≠, <, or >. | Hₐ: μ ≠ μ₀ (two-tailed) Hₐ: μ > μ₀ (right) Hₐ: μ < μ₀ (left) |
Types of Tests
| Test Type | Alternative Hypothesis | When to Use |
|---|---|---|
| Two-Tailed | Hₐ: parameter ≠ value | Testing for any difference (higher OR lower) |
| Right-Tailed | Hₐ: parameter > value | Testing if parameter is greater than claimed |
| Left-Tailed | Hₐ: parameter < value | Testing if parameter is less than claimed |
Components of Hypothesis Testing
- Test Statistic: Value calculated from sample data measuring how far sample is from hypothesized parameter
- Significance Level (α): Probability threshold for rejecting H₀ (commonly 0.05, 0.01, or 0.10)
- Critical Value: Boundary of rejection region
- P-value: Probability of obtaining results as extreme as observed, if H₀ is true
Steps of Hypothesis Testing
- State H₀ and Hₐ, choose α
- Calculate test statistic from sample data
- Find critical value(s) or p-value
- Make decision: reject or fail to reject H₀
- State conclusion in context
- Critical Value Approach: Reject H₀ if test statistic falls in critical region
- P-value Approach: Reject H₀ if p-value ≤ α
2. Type I and Type II Errors & Power
The Four Possible Outcomes
| Our Decision | H₀ is True | H₀ is False |
|---|---|---|
| Reject H₀ | Type I Error (α) False Positive |
Correct Decision Power (1 - β) |
| Fail to Reject H₀ | Correct Decision (1 - α) |
Type II Error (β) False Negative |
Type I Error (α)
Definition: Rejecting a true null hypothesis (false positive)
Probability: α (significance level we choose)
Example: Conclude a drug works when it actually doesn't
Control: We directly control by choosing α (0.05, 0.01, etc.)
Type II Error (β)
Definition: Failing to reject a false null hypothesis (false negative)
Probability: β (depends on sample size, effect size, α)
Example: Conclude a drug doesn't work when it actually does
Control: Indirectly controlled by increasing sample size
Statistical Power
Probability of correctly rejecting a false H₀
Recommended: Power ≥ 0.80 (80%)
Factors Affecting Power
| Factor | To Increase Power | Effect |
|---|---|---|
| Sample Size (n) | Increase n | Most practical method |
| Significance Level (α) | Increase α | But increases Type I error |
| Effect Size | Larger effect | Not always controllable |
| Variability (σ) | Decrease σ | Better measurements |
3. Hypothesis Tests for Means
Z-Test vs. T-Test
| Test | When to Use | Test Statistic |
|---|---|---|
| Z-Test | σ (population SD) is KNOWN | z = (x̄ - μ₀) / (σ/√n) |
| T-Test | σ is UNKNOWN (use s) | t = (x̄ - μ₀) / (s/√n) df = n - 1 |
Z-Test for Population Mean
Where: x̄ = sample mean, μ₀ = hypothesized mean, σ = population SD, n = sample size
T-Test for Population Mean
Where: s = sample SD, df = n - 1
Use t-distribution table with df to find critical values
Critical Values (Standard Normal Distribution)
| Test Type | α = 0.10 | α = 0.05 | α = 0.01 |
|---|---|---|---|
| Two-tailed | ±1.645 | ±1.96 | ±2.576 |
| Right-tailed | 1.282 | 1.645 | 2.326 |
| Left-tailed | -1.282 | -1.645 | -2.326 |
Decision Rules for Tests of Means
| Test Type | Reject H₀ if... |
|---|---|
| Two-tailed | |z| > critical value or |t| > critical value |
| Right-tailed | z > critical value or t > critical value |
| Left-tailed | z < critical value or t < critical value |
4. Hypothesis Tests for Proportions
When to Use
Use proportion tests when data is categorical (yes/no, success/failure) and can be expressed as a proportion or percentage.
Notation
| Symbol | Meaning |
|---|---|
| p | Population proportion (parameter, unknown) |
| p₀ | Hypothesized population proportion (in H₀) |
| p̂ | Sample proportion = x/n |
| x | Number of successes in sample |
| n | Sample size |
Test Statistic for Proportions
Important: Use p₀ in denominator, NOT p̂
Required Conditions
Before conducting test, verify:
- Random sample
- Independent observations
- Normal approximation: np₀ ≥ 10 AND n(1-p₀) ≥ 10
If conditions fail, test is not valid!
5. Common Mistakes to Avoid
Don't Make These Errors!
- Wrong hypotheses: H₀ always has =, never ≥ or ≤
- Confusing α and p-value: α is chosen before test; p-value is calculated from data
- Saying "accept H₀": We "fail to reject" H₀, never "accept" it
- Using p̂ in proportion SE: For hypothesis tests, use p₀ in √(p₀(1-p₀)/n)
- Ignoring conditions: Always check conditions before conducting test
- Wrong test choice: Use t-test when σ is unknown (almost always)
- Misinterpreting p-value: p-value ≠ P(H₀ is true)
- Confusing statistical and practical significance: Significant ≠ important
6. Formula Summary
| Test | Formula | Notes |
|---|---|---|
| Z-Test (Mean) | z = (x̄ - μ₀) / (σ/√n) | When σ is known |
| T-Test (Mean) | t = (x̄ - μ₀) / (s/√n) df = n - 1 |
When σ is unknown |
| Z-Test (Proportion) | z = (p̂ - p₀) / √(p₀(1-p₀)/n) | Check: np₀ ≥ 10, n(1-p₀) ≥ 10 |
| Sample Proportion | p̂ = x/n | x = successes, n = sample size |
| Power | Power = 1 - β | β = P(Type II error) |
7. Interpreting Results
P-Value Interpretation
| P-value Range | Interpretation |
|---|---|
| p < 0.01 | Very strong evidence against H₀ |
| 0.01 ≤ p < 0.05 | Strong evidence against H₀ |
| 0.05 ≤ p < 0.10 | Weak evidence against H₀ |
| p ≥ 0.10 | Little or no evidence against H₀ |
Conclusion Language
If we reject H₀:
"There is sufficient evidence at the α = ___ significance level to [conclude Hₐ in context]."
If we fail to reject H₀:
"There is not sufficient evidence at the α = ___ significance level to [conclude Hₐ in context]."
Study Tips
- Practice setting up hypotheses—this is crucial for everything else
- Make a flowchart for choosing z-test vs t-test
- Always check conditions before running a test
- Memorize common critical values (±1.96 for α = 0.05, two-tailed)
- Practice interpreting results in context, not just mathematically
- Understand the difference between Type I and Type II errors—use real-world examples