Statistics 101: Hypothesis Testing & Inferential Statistics
Complete Study Guide with Examples & Solutions
Table of Contents
1. Introduction to Hypothesis Testing
2. Types of Hypothesis Tests
3. One-Sample Tests
4. Two-Sample Tests
5. Chi-Square Tests
6. ANOVA (Analysis of Variance)
7. Correlation and Regression
8. Step-by-Step Problem Solutions
9. Statistical Tables & Formulas
10. Common Exam Questions
Introduction to Hypothesis Testing
Key Concepts
Null Hypothesis (H₀): The statement being tested, usually stating "no effect" or "no difference"
Alternative Hypothesis (H₁ or Hₐ): The statement we're trying to prove
Types of Tests
Two-tailed: H₁: μ ≠ μ₀ (testing for any difference)
Left-tailed: H₁: μ < μ₀ (testing if less than)
Right-tailed: H₁: μ > μ₀ (testing if greater than)
Decision Making Process
1. State hypotheses (H₀ and H₁)
2. Choose significance level (α), typically 0.05
3. Calculate test statistic
4. Find p-value or critical value
5. Make decision: Reject H₀ if p-value < α
6. State conclusion in context
, Type I and Type II Errors
Type I Error (α): Rejecting true H₀ (False Positive)
Type II Error (β): Failing to reject false H₀ (False Negative)
Power = 1 - β: Probability of correctly rejecting false H₀
Types of Hypothesis Tests
Choosing the Right Test
Data Type Sample Size Known σ Test to Use
Quantitative n ≥ 30 Yes Z-test
Quantitative n < 30 No t-test
Categorical Any N/A Chi-square
Two means Large samples Yes Two-sample Z
Two means Small samples No Two-sample t
One-Sample Tests
One-Sample Z-Test
When to use: Testing one mean when σ is known and n ≥ 30
Test Statistic:
z = (x̄ - μ₀)/(σ/√n)
Example Problem: A manufacturer claims light bulbs last 1000 hours on average. A sample of 36 bulbs
has mean life 980 hours with known σ = 120. Test at α = 0.05.
Solution:
1. H₀: μ = 1000, H₁: μ ≠ 1000 (two-tailed)
2. α = 0.05
3. z = (980 - 1000)/(120/√36) = -20/20 = -1.00
4. p-value = 2 × P(Z < -1.00) = 2 × 0.1587 = 0.3174
5. Since 0.3174 > 0.05, fail to reject H₀
6. Insufficient evidence that mean differs from 1000 hours
Complete Study Guide with Examples & Solutions
Table of Contents
1. Introduction to Hypothesis Testing
2. Types of Hypothesis Tests
3. One-Sample Tests
4. Two-Sample Tests
5. Chi-Square Tests
6. ANOVA (Analysis of Variance)
7. Correlation and Regression
8. Step-by-Step Problem Solutions
9. Statistical Tables & Formulas
10. Common Exam Questions
Introduction to Hypothesis Testing
Key Concepts
Null Hypothesis (H₀): The statement being tested, usually stating "no effect" or "no difference"
Alternative Hypothesis (H₁ or Hₐ): The statement we're trying to prove
Types of Tests
Two-tailed: H₁: μ ≠ μ₀ (testing for any difference)
Left-tailed: H₁: μ < μ₀ (testing if less than)
Right-tailed: H₁: μ > μ₀ (testing if greater than)
Decision Making Process
1. State hypotheses (H₀ and H₁)
2. Choose significance level (α), typically 0.05
3. Calculate test statistic
4. Find p-value or critical value
5. Make decision: Reject H₀ if p-value < α
6. State conclusion in context
, Type I and Type II Errors
Type I Error (α): Rejecting true H₀ (False Positive)
Type II Error (β): Failing to reject false H₀ (False Negative)
Power = 1 - β: Probability of correctly rejecting false H₀
Types of Hypothesis Tests
Choosing the Right Test
Data Type Sample Size Known σ Test to Use
Quantitative n ≥ 30 Yes Z-test
Quantitative n < 30 No t-test
Categorical Any N/A Chi-square
Two means Large samples Yes Two-sample Z
Two means Small samples No Two-sample t
One-Sample Tests
One-Sample Z-Test
When to use: Testing one mean when σ is known and n ≥ 30
Test Statistic:
z = (x̄ - μ₀)/(σ/√n)
Example Problem: A manufacturer claims light bulbs last 1000 hours on average. A sample of 36 bulbs
has mean life 980 hours with known σ = 120. Test at α = 0.05.
Solution:
1. H₀: μ = 1000, H₁: μ ≠ 1000 (two-tailed)
2. α = 0.05
3. z = (980 - 1000)/(120/√36) = -20/20 = -1.00
4. p-value = 2 × P(Z < -1.00) = 2 × 0.1587 = 0.3174
5. Since 0.3174 > 0.05, fail to reject H₀
6. Insufficient evidence that mean differs from 1000 hours
