2026
STATISTICS II
SUMMARY
KOEN HANEGREEFS
VUB
,Table of Contents
Statistics II — Course Summary ...................................................................................... 5
How to use this summary ............................................................................................ 5
Notation conventions ................................................................................................. 5
Common critical values (memorise!) ........................................................................... 6
Chapter 0 — Course Introduction & Statistics I Recap ...................................................... 7
Overview (Neutral) ...................................................................................................... 7
Course Structure (Less important) ............................................................................... 7
Course Content Topics (Neutral) ................................................................................. 7
What You Need From Statistics I (Important) ................................................................ 7
Chapter 10 — Sampling Distributions and Confidence Intervals for Proportions ................ 9
10.1 The Sampling Distribution of a Sample Proportion (Important) ............................... 9
10.2 Assumptions and Conditions for the Normal Model (Important)............................. 9
10.3 The 68–95–99.7 Rule Applied to 𝑝 (Important) ..................................................... 10
10.4 The Standard Error (Important) ........................................................................... 11
10.5 Confidence Interval for a Proportion (Important) ................................................. 11
10.6 Interpreting a Confidence Interval (Important — Common Exam Pitfall) ............... 12
10.7 Margin of Error: Certainty vs. Precision (Important) ............................................. 13
10.8 Choosing the Sample Size (Important) ................................................................ 13
10.9 Common Errors and Pitfalls (Important) ............................................................. 14
10.10 One-Proportion z-Test Preview (Neutral) ........................................................... 14
10.11 Summary of Key Formulas (Important) .............................................................. 15
Chapter 11 — Confidence Intervals for Means ............................................................... 16
11.1 From Proportions to Means (Neutral) .................................................................. 16
11.2 The Sampling Distribution of the Sample Mean (Important) ................................. 16
11.3 Why We Need the t-Distribution (Important) ....................................................... 17
11.4 Student’s t-Distribution (Important) ................................................................... 18
11.5 One-Sample t Confidence Interval (Important) .................................................... 19
11.6 Assumptions and Conditions (Important) ........................................................... 20
11.7 Interpretation of a Confidence Interval (Important) ............................................. 21
11.8 Sample Size Determination (Important) .............................................................. 21
11.9 z-Interval vs t-Interval: Brief Comparison (Neutral) .............................................. 22
11.10 Bootstrap Confidence Intervals (Less important) .............................................. 22
Koen Hanegreefs 1
, 11.12 Summary of Key Formulas (Important) .............................................................. 23
Chapter 12 — Testing Hypotheses About Proportions .................................................... 24
Overview .................................................................................................................. 24
12.1 The Logic of Hypothesis Testing (Important) ........................................................ 24
12.2 Setting Up Hypotheses: 𝐻0 and 𝐻𝐴 (Important) .................................................. 24
12.3 The One-Proportion z-Test (Important) ............................................................... 25
12.4 The p-Value: Definition and Interpretation (Important)......................................... 26
12.5 Decision Rule: Significance Level 𝛼 (Important) .................................................. 27
12.6 Conclusion: In Words, In Context (Important) ..................................................... 27
12.7 Connection Between Confidence Intervals and Two-Sided Tests (Neutral) ........... 28
12.8 Critical Value Approach vs p-Value Approach (Neutral) ....................................... 28
12.9 Common Pitfalls and Misinterpretations (Important) ........................................... 29
12.10 Full Worked Example: Home Advantage in Baseball (Neutral) ............................ 29
Summary Table: One-Proportion z-Test ..................................................................... 30
Chapter 13 — More About Tests & Intervals ................................................................... 32
13.1 The One-Sample t-Test for the Mean (Important) ................................................. 32
13.2 The One-Sample t-Confidence Interval (Recap from Ch. 11) (Important) .............. 33
13.3 P-Values and the t-Distribution (Important) ........................................................ 33
13.4 Two-Sided vs. One-Sided Alternatives (Important) .............................................. 34
13.5 Connection Between Confidence Intervals and Two-Sided Tests (Important) ....... 34
13.6 The Paired t-Test (Matched-Pairs Design) (Important) .......................................... 35
13.7 Alpha Level and Significance (Important) ............................................................ 36
13.8 Type I and Type II Errors (Important) .................................................................... 36
13.9 Effect Size and Statistical vs. Practical Significance (Important) .......................... 37
13.10 Power = 1 − 𝛽 (Important) ................................................................................ 38
13.11 Common Pitfalls (Important) ............................................................................ 39
13.12 Formula Reference Table (Important) ............................................................... 39
Chapter 14 — Comparing Two Means ............................................................................ 41
14.1 Two Independent Samples: Comparing 𝜇1 − 𝜇2 (Important) ................................ 41
14.2 The Two-Sample 𝑡-Test (Welch / Unpooled) (Important) ...................................... 41
14.3 Two-Sample 𝑡 Confidence Interval for 𝜇1 − 𝜇2 (Important) .................................. 43
14.4 The Pooled 𝑡-Test (Neutral) ................................................................................ 43
14.5 Conditions and Assumptions (Important) ........................................................... 44
Koen Hanegreefs 2
, 14.6 Two-Proportion 𝑧-Test and CI: Comparing 𝑝1 − 𝑝2 (Important) ............................ 45
14.7 Paired vs. Independent Two-Sample: When Each Applies (Important) .................. 45
14.8 Common Pitfalls (Important).............................................................................. 47
14.9 Formula Summary Table .................................................................................... 47
14.11 Decision Guide — Which Test to Use? .............................................................. 47
Chapter 15 — Inference for Counts: Chi-Square Tests ................................................... 49
15.1 When to Use Chi-Square Tests (Important) ......................................................... 49
15.2 The Three Chi-Square Tests (Important) .............................................................. 49
15.3 The Chi-Square Test Statistic (Important) ........................................................... 51
15.4 Computing Expected Counts (Important) ........................................................... 51
15.5 Conditions for Validity (Important)...................................................................... 52
15.6 Standardized Residuals — Which Cells Drive Significance (Important) ................. 52
15.7 Relationship between Chi-Square (2 × 2) and the Two-Proportion z-Test (Important)
................................................................................................................................ 53
15.8 Reading and Interpreting R Output (Important) .................................................... 53
15.9 Step-by-Step Procedure for Any Chi-Square Test (Important)............................... 54
15.10 Common Pitfalls (Important) ............................................................................ 54
15.11 Summary Table (Important).............................................................................. 55
Chapter 16 — Linear Regression and Inference for Regression ....................................... 56
16.1 Recap: Correlation (Chapter 4) (Important) ......................................................... 56
16.2 The Importance of Making Graphs — Anscombe’s Quartet (Important) ................ 57
16.3 The Linear Regression Model (Important) ............................................................ 57
16.4 Interpreting Slope and Intercept (Important) ....................................................... 58
16.5 Understanding Regression from Correlation (Neutral) ......................................... 58
16.6 Coefficient of Determination 𝑅2 (Important) ....................................................... 59
16.7 Standard Deviation of Residuals 𝑠𝑒 (Important)................................................... 59
16.8 Assumptions and Conditions (LINE) (Important) ................................................. 59
16.9 Outliers, Leverage, and Influential Points (Important) .......................................... 60
16.10 Extrapolation Warning (Neutral) ....................................................................... 61
16.11 Inference for Regression (Chapter 16) (Important) ............................................. 61
16.12 Inference for Predicted Values (Important)........................................................ 62
16.13 Reading R lm() Output (Important) .................................................................. 63
16.14 Summary Table: Inference in Simple Linear Regression (Important) ................... 64
Koen Hanegreefs 3
STATISTICS II
SUMMARY
KOEN HANEGREEFS
VUB
,Table of Contents
Statistics II — Course Summary ...................................................................................... 5
How to use this summary ............................................................................................ 5
Notation conventions ................................................................................................. 5
Common critical values (memorise!) ........................................................................... 6
Chapter 0 — Course Introduction & Statistics I Recap ...................................................... 7
Overview (Neutral) ...................................................................................................... 7
Course Structure (Less important) ............................................................................... 7
Course Content Topics (Neutral) ................................................................................. 7
What You Need From Statistics I (Important) ................................................................ 7
Chapter 10 — Sampling Distributions and Confidence Intervals for Proportions ................ 9
10.1 The Sampling Distribution of a Sample Proportion (Important) ............................... 9
10.2 Assumptions and Conditions for the Normal Model (Important)............................. 9
10.3 The 68–95–99.7 Rule Applied to 𝑝 (Important) ..................................................... 10
10.4 The Standard Error (Important) ........................................................................... 11
10.5 Confidence Interval for a Proportion (Important) ................................................. 11
10.6 Interpreting a Confidence Interval (Important — Common Exam Pitfall) ............... 12
10.7 Margin of Error: Certainty vs. Precision (Important) ............................................. 13
10.8 Choosing the Sample Size (Important) ................................................................ 13
10.9 Common Errors and Pitfalls (Important) ............................................................. 14
10.10 One-Proportion z-Test Preview (Neutral) ........................................................... 14
10.11 Summary of Key Formulas (Important) .............................................................. 15
Chapter 11 — Confidence Intervals for Means ............................................................... 16
11.1 From Proportions to Means (Neutral) .................................................................. 16
11.2 The Sampling Distribution of the Sample Mean (Important) ................................. 16
11.3 Why We Need the t-Distribution (Important) ....................................................... 17
11.4 Student’s t-Distribution (Important) ................................................................... 18
11.5 One-Sample t Confidence Interval (Important) .................................................... 19
11.6 Assumptions and Conditions (Important) ........................................................... 20
11.7 Interpretation of a Confidence Interval (Important) ............................................. 21
11.8 Sample Size Determination (Important) .............................................................. 21
11.9 z-Interval vs t-Interval: Brief Comparison (Neutral) .............................................. 22
11.10 Bootstrap Confidence Intervals (Less important) .............................................. 22
Koen Hanegreefs 1
, 11.12 Summary of Key Formulas (Important) .............................................................. 23
Chapter 12 — Testing Hypotheses About Proportions .................................................... 24
Overview .................................................................................................................. 24
12.1 The Logic of Hypothesis Testing (Important) ........................................................ 24
12.2 Setting Up Hypotheses: 𝐻0 and 𝐻𝐴 (Important) .................................................. 24
12.3 The One-Proportion z-Test (Important) ............................................................... 25
12.4 The p-Value: Definition and Interpretation (Important)......................................... 26
12.5 Decision Rule: Significance Level 𝛼 (Important) .................................................. 27
12.6 Conclusion: In Words, In Context (Important) ..................................................... 27
12.7 Connection Between Confidence Intervals and Two-Sided Tests (Neutral) ........... 28
12.8 Critical Value Approach vs p-Value Approach (Neutral) ....................................... 28
12.9 Common Pitfalls and Misinterpretations (Important) ........................................... 29
12.10 Full Worked Example: Home Advantage in Baseball (Neutral) ............................ 29
Summary Table: One-Proportion z-Test ..................................................................... 30
Chapter 13 — More About Tests & Intervals ................................................................... 32
13.1 The One-Sample t-Test for the Mean (Important) ................................................. 32
13.2 The One-Sample t-Confidence Interval (Recap from Ch. 11) (Important) .............. 33
13.3 P-Values and the t-Distribution (Important) ........................................................ 33
13.4 Two-Sided vs. One-Sided Alternatives (Important) .............................................. 34
13.5 Connection Between Confidence Intervals and Two-Sided Tests (Important) ....... 34
13.6 The Paired t-Test (Matched-Pairs Design) (Important) .......................................... 35
13.7 Alpha Level and Significance (Important) ............................................................ 36
13.8 Type I and Type II Errors (Important) .................................................................... 36
13.9 Effect Size and Statistical vs. Practical Significance (Important) .......................... 37
13.10 Power = 1 − 𝛽 (Important) ................................................................................ 38
13.11 Common Pitfalls (Important) ............................................................................ 39
13.12 Formula Reference Table (Important) ............................................................... 39
Chapter 14 — Comparing Two Means ............................................................................ 41
14.1 Two Independent Samples: Comparing 𝜇1 − 𝜇2 (Important) ................................ 41
14.2 The Two-Sample 𝑡-Test (Welch / Unpooled) (Important) ...................................... 41
14.3 Two-Sample 𝑡 Confidence Interval for 𝜇1 − 𝜇2 (Important) .................................. 43
14.4 The Pooled 𝑡-Test (Neutral) ................................................................................ 43
14.5 Conditions and Assumptions (Important) ........................................................... 44
Koen Hanegreefs 2
, 14.6 Two-Proportion 𝑧-Test and CI: Comparing 𝑝1 − 𝑝2 (Important) ............................ 45
14.7 Paired vs. Independent Two-Sample: When Each Applies (Important) .................. 45
14.8 Common Pitfalls (Important).............................................................................. 47
14.9 Formula Summary Table .................................................................................... 47
14.11 Decision Guide — Which Test to Use? .............................................................. 47
Chapter 15 — Inference for Counts: Chi-Square Tests ................................................... 49
15.1 When to Use Chi-Square Tests (Important) ......................................................... 49
15.2 The Three Chi-Square Tests (Important) .............................................................. 49
15.3 The Chi-Square Test Statistic (Important) ........................................................... 51
15.4 Computing Expected Counts (Important) ........................................................... 51
15.5 Conditions for Validity (Important)...................................................................... 52
15.6 Standardized Residuals — Which Cells Drive Significance (Important) ................. 52
15.7 Relationship between Chi-Square (2 × 2) and the Two-Proportion z-Test (Important)
................................................................................................................................ 53
15.8 Reading and Interpreting R Output (Important) .................................................... 53
15.9 Step-by-Step Procedure for Any Chi-Square Test (Important)............................... 54
15.10 Common Pitfalls (Important) ............................................................................ 54
15.11 Summary Table (Important).............................................................................. 55
Chapter 16 — Linear Regression and Inference for Regression ....................................... 56
16.1 Recap: Correlation (Chapter 4) (Important) ......................................................... 56
16.2 The Importance of Making Graphs — Anscombe’s Quartet (Important) ................ 57
16.3 The Linear Regression Model (Important) ............................................................ 57
16.4 Interpreting Slope and Intercept (Important) ....................................................... 58
16.5 Understanding Regression from Correlation (Neutral) ......................................... 58
16.6 Coefficient of Determination 𝑅2 (Important) ....................................................... 59
16.7 Standard Deviation of Residuals 𝑠𝑒 (Important)................................................... 59
16.8 Assumptions and Conditions (LINE) (Important) ................................................. 59
16.9 Outliers, Leverage, and Influential Points (Important) .......................................... 60
16.10 Extrapolation Warning (Neutral) ....................................................................... 61
16.11 Inference for Regression (Chapter 16) (Important) ............................................. 61
16.12 Inference for Predicted Values (Important)........................................................ 62
16.13 Reading R lm() Output (Important) .................................................................. 63
16.14 Summary Table: Inference in Simple Linear Regression (Important) ................... 64
Koen Hanegreefs 3
