2026
STATISTICS II
SUMMARY WITH TI84 GUIDE
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 ...................................................... 6
Overview (Neutral) ...................................................................................................... 6
Course Structure (Less important) ............................................................................... 6
Course Content Topics (Neutral) ................................................................................. 7
What You Need From Statistics I (Important) ................................................................ 7
Chapter 10 — Sampling Distributions and Confidence Intervals for Proportions ................ 8
10.1 The Sampling Distribution of a Sample Proportion (Important) ............................... 8
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) ........................................................................... 10
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) ............................................. 12
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 ............................................................... 15
11.1 From Proportions to Means (Neutral) .................................................................. 15
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) ................................................................... 17
11.5 One-Sample t Confidence Interval (Important) .................................................... 18
11.6 Assumptions and Conditions (Important) ........................................................... 19
11.7 Interpretation of a Confidence Interval (Important) ............................................. 20
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
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)......................................... 27
12.5 Decision Rule: Significance Level 𝛼 (Important) .................................................. 27
12.6 Conclusion: In Words, In Context (Important) ..................................................... 28
12.7 Connection Between Confidence Intervals and Two-Sided Tests (Neutral) ........... 28
12.8 Critical Value Approach vs p-Value Approach (Neutral) ....................................... 29
12.9 Common Pitfalls and Misinterpretations (Important) ........................................... 30
12.10 Full Worked Example: Home Advantage in Baseball (Neutral) ............................ 30
Summary Table: One-Proportion z-Test ..................................................................... 31
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) .................................................................... 37
13.9 Effect Size and Statistical vs. Practical Significance (Important) .......................... 38
13.10 Power = 1 − 𝛽 (Important) ................................................................................ 38
13.11 Common Pitfalls (Important) ............................................................................ 39
13.12 Formula Reference Table (Important) ............................................................... 40
Chapter 14 — Comparing Two Means ............................................................................ 40
14.1 Two Independent Samples: Comparing 𝜇1 − 𝜇2 (Important) ................................ 40
14.2 The Two-Sample 𝑡-Test (Welch / Unpooled) (Important) ...................................... 41
14.3 Two-Sample 𝑡 Confidence Interval for 𝜇1 − 𝜇2 (Important) .................................. 42
14.4 The Pooled 𝑡-Test (Neutral) ................................................................................ 43
14.5 Conditions and Assumptions (Important) ........................................................... 44
2
, 14.6 Two-Proportion 𝑧-Test and CI: Comparing 𝑝1 − 𝑝2 (Important) ............................ 44
14.7 Paired vs. Independent Two-Sample: When Each Applies (Important) .................. 45
14.8 Common Pitfalls (Important).............................................................................. 46
14.9 Formula Summary Table .................................................................................... 47
14.11 Decision Guide — Which Test to Use? .............................................................. 47
Chapter 15 — Inference for Counts: Chi-Square Tests ................................................... 47
15.1 When to Use Chi-Square Tests (Important) ......................................................... 47
15.2 The Three Chi-Square Tests (Important) .............................................................. 48
15.3 The Chi-Square Test Statistic (Important) ........................................................... 49
15.4 Computing Expected Counts (Important) ........................................................... 50
15.5 Conditions for Validity (Important)...................................................................... 51
15.6 Standardized Residuals — Which Cells Drive Significance (Important) ................. 51
15.7 Relationship between Chi-Square (2 × 2) and the Two-Proportion z-Test (Important)
................................................................................................................................ 52
15.8 Running and Reading the Chi-Square Test on the TI-84 (Important) ...................... 52
15.9 Step-by-Step Procedure for Any Chi-Square Test (Important)............................... 53
15.10 Common Pitfalls (Important) ............................................................................ 53
15.11 Summary Table (Important).............................................................................. 54
Chapter 16 — Linear Regression and Inference for Regression ....................................... 54
16.1 Recap: Correlation (Chapter 4) (Important) ......................................................... 54
16.2 The Importance of Making Graphs — Anscombe’s Quartet (Important) ................ 55
16.3 The Linear Regression Model (Important) ............................................................ 56
16.4 Interpreting Slope and Intercept (Important) ....................................................... 57
16.5 Understanding Regression from Correlation (Neutral) ......................................... 57
16.6 Coefficient of Determination 𝑅2 (Important) ....................................................... 57
16.7 Standard Deviation of Residuals 𝑠𝑒 (Important)................................................... 58
16.8 Assumptions and Conditions (LINE) (Important) ................................................. 58
16.9 Outliers, Leverage, and Influential Points (Important) .......................................... 59
16.10 Extrapolation Warning (Neutral) ....................................................................... 60
16.11 Inference for Regression (Chapter 16) (Important) ............................................. 60
16.12 Inference for Predicted Values (Important)........................................................ 61
16.13 Reading TI-84 LinRegTTest Output (Important) ............................................... 62
16.14 Summary Table: Inference in Simple Linear Regression (Important) ................... 63
3
, 16.16 Common Pitfalls and What Can Go Wrong (Important) ...................................... 64
16.17 Quick Reference: Key Formulas (Important) ...................................................... 64
Chapter 17 — Inference for Regression and Residual Analysis ........................................ 65
17.1 The Population Regression Model (Important) ..................................................... 65
17.2 Sampling Distribution of the Slope 𝑏1 (Important) ............................................... 66
17.3 Hypothesis Test for the Slope (Important) ........................................................... 66
17.4 Confidence Interval for the Slope (Important) ..................................................... 67
17.5 Reading TI-84 LinRegTTest Output (Important) ................................................. 67
17.6 Standard Errors for Predicted Values (Important) ................................................ 68
17.7 Conditions for Inference — The LINE Assumptions (Important) ............................ 69
17.8 Unusual and Extraordinary Observations (Important) .......................................... 71
17.9 Extrapolation (Neutral) ...................................................................................... 72
17.10 Working with Summarized Data (Neutral) ......................................................... 72
17.11 Data Transformations (Important) .................................................................... 72
17.12 Overview of Inference for Regression ................................................................ 73
Chapter 23 — Non-Parametric Methods ........................................................................ 73
23.1 Why Non-Parametric Methods? (Important)........................................................ 73
23.2 Wilcoxon Rank-Sum / Mann–Whitney Test (Important) ........................................ 74
23.3 Kruskal-Wallis Test (Important) .......................................................................... 76
23.4 Wilcoxon Signed-Rank Test (Important) .............................................................. 76
23.5 Kendall’s Tau — Measuring Monotonicity (Neutral) ............................................. 78
23.6 Spearman’s Rank Correlation 𝜌 (Neutral) ........................................................... 78
23.7 Conditions Checklist Summary (Important) ........................................................ 79
Course Wrap-Up & Method Selector ............................................................................. 80
W.1 The Decision Framework (Important) .................................................................. 80
W.2 Decision Tree ..................................................................................................... 80
W.3 Complete Method Cheat-Sheet (Important) ........................................................ 82
W.4 Conditions Checklist for Each Test (Important) .................................................... 84
W.5 The LEGO Example — Applying the Selector (Important) ...................................... 84
W.6 Common Exam Pitfalls (Important) ..................................................................... 85
W.7 Full Exam Formula Reference (Non-Parametric) (Important) ................................ 86
4
,Statistics II — Course Summary
Course: Statistics for Business and Economics II (1009724BNR) — Vrije Universiteit
Brussel (VUB) Handbook: Business Statistics, 4th global edition — Sharpe, De Veaux &
Velleman
How to use this summary
This is the theory-only companion to Statistics II — Worked Solutions Manual. It distils all
course content (HOC1–HOC11 + LEGO recap docs) into roughly 55 pages, structured by
chapter. Worked exercises live in the separate Solutions Manual; pair the two documents
while studying.
Every subsection is tagged with a priority label so you can revise efficiently:
Tag Meaning What to do
(Important) Core exam material — high probability Study deeply, memorise
of appearing formulas
(Neutral) Standard course content — supports Read once carefully;
the Important parts understand the logic
(Less Side topics, advanced caveats, course Skim for awareness
important) logistics
The exam is closed-book multiple-choice (20 questions, 50 % theory + 50 %
exercises) with a higher-pass-mark scheme. The formula sheet and tables from
Canvas are provided.
Notation conventions
Symbol Meaning
𝜇, 𝜎, 𝑝 Population parameters (unknown)
𝑦‾, 𝑠, 𝑝̂ Sample statistics (estimates)
𝑛 Sample size
∗ Critical value from the standard Normal distribution
𝑧
∗
𝑡"# Critical value from the 𝑡-distribution with 𝑑𝑓 degrees of freedom
𝑆𝐸 Standard Error of a statistic
𝐻$ , 𝐻% Null hypothesis, alternative hypothesis
𝛼 Significance level
𝑝-value Probability of seeing data at least as extreme as observed under 𝐻$
𝛽$ , 𝛽& Population regression intercept and slope
5
,𝑏$ , 𝑏& Sample regression intercept and slope
𝑦< Predicted (fitted) response from a regression model
𝑒 Residual (𝑦 − 𝑦<)
Common critical values (memorise!)
Confidence level Two-sided 𝑧 ∗ Right-tail 𝑧 ∗ (𝛼)
90 % 1.645 𝑧 ∗ = 1.282 for 𝛼 = 0.10
95 % 1.96 𝑧 ∗ = 1.645 for 𝛼 = 0.05
98 % 2.33 𝑧 ∗ = 2.054 for 𝛼 = 0.02
99 % 2.576 𝑧 ∗ = 2.326 for 𝛼 = 0.01
For 𝑡, the critical value depends on 𝑑𝑓 — read from the 𝑡-table (provided at the exam).
Chapter 0 — Course Introduction & Statistics I Recap
Overview (Neutral)
Statistics for Business and Economics II (course code 1009724BNR) is the follow-up to
Statistics I, using the handbook Business Statistics, 4th global edition, by Sharpe, De
Veaux & Velleman (3rd edition is also usable — a comparison table is on Canvas). The
course runs from February 2026. The lecturer is Evy Rombaut; seminars are led by Dennis
Verbist.
Course Structure (Less important)
Component Details
Lectures Monday 1–4 pm; theory + LEGO examples
Seminars Tuesday or Friday; exercises by hand and on the TI-84
Statistics Café Wednesday 2–4 pm; optional Q&A/drop-in
Evaluation:
• 10 % — intermediate test (on-campus, ~May 11; automatically transferred to 2nd
session — no retake)
• 90 % — multiple-choice exam: 50 % theory + 50 % exercises
– Must score ≥ 10/20 on the exam to pass the course
– Retake in 2nd session is possible if total score < 10/20
6
,TI-84 on the exam: You must be able to (1) enter data into lists (or a matrix for chi-square),
(2) run the correct test/command from STAT ▸ TESTS or the DISTR menu, and (3) interpret
the output.
Course Content Topics (Neutral)
Module Handbook chapters
Sampling Chapter 10
Hypothesis testing Chapters 11–15
Correlation & Regression Chapters 4, 16–17
Non-parametric Methods Chapter 23
What You Need From Statistics I (Important)
The following concepts are assumed known. They are not re-taught but are used
constantly throughout the course.
Parameters vs. Statistics (Important)
• Parameter — a numerical summary of the population (fixed but usually unknown);
denoted with Greek letters, e.g. 𝜇, 𝜎, 𝑝.
• Statistic — a numerical summary computed from a sample; denoted with Roman
letters or hats, e.g. 𝑥‾, 𝑠, 𝑝̂ .
Population vs. Sample (Important)
• Population — the entire group you want to draw conclusions about.
• Sample — a subset of the population, studied because surveying the whole
population is infeasible.
Descriptive vs. Inferential Statistics (Important)
Type Goal
Descriptive Summarise and describe the data you have (histograms, means, proportions)
Inferential Draw conclusions about the population based on a sample
Statistics II is almost entirely about inferential statistics.
Types of Variables (Important)
• Categorical (qualitative) — values are labels/categories; summarised by
proportions/counts.
– Binary/dichotomous: only two possible values (e.g. success/failure, yes/no).
• Quantitative (numerical) — values are numbers; summarised by means, standard
deviations.
7
, – Discrete: countable integer values.
– Continuous: any value on an interval.
The distinction matters because Chapter 10 deals exclusively with categorical variables
and proportions, while later chapters deal with means.
Key Distributions to Know (Important)
• Normal distribution 𝑁(𝜇, 𝜎) — bell-shaped, symmetric; characterised by the 68–95–
99.7 rule.
• Standard Normal 𝑁(0,1) — used for z-scores.
'()
• z-score: 𝑧 = *
Concept of Sampling Variability (Important)
Different samples yield different statistics. The sampling distribution is the distribution of
a sample statistic across all possible samples — understanding this is central to the entire
course.
Chapter 10 — Sampling Distributions and Confidence Intervals for
Proportions
10.1 The Sampling Distribution of a Sample Proportion (Important)
Setup
We have a population in which a proportion 𝑝 of individuals have some characteristic
(“success”). We draw a random sample of size 𝑛 and observe 𝑝̂ , the sample proportion:
number of successes in sample
𝑝̂ =
𝑛
𝑝 is the unknown parameter; 𝑝̂ is the statistic we observe.
The Sampling Distribution
Repeated sampling yields a sampling distribution of the proportion — bell-shaped and
centred at 𝑝 (verified by simulation). Under the right conditions:
𝑝𝑞
𝑝̂ ∼ 𝑁 I𝑝, K M, 𝑞 =1−𝑝
𝑛
8
, Property Value
Centre (mean) 𝑝
Standard deviation 𝑝𝑞
𝑆𝐷(𝑝̂ ) = K
𝑛
Shape Approximately Normal (when conditions are met)
Key insight: even though we can only draw one sample in practice, we can use the known
sampling distribution model to make probability statements about where 𝑝̂ will fall relative
to 𝑝.
10.2 Assumptions and Conditions for the Normal Model (Important)
Two assumptions must hold, each checked via conditions:
Assumption 1 — Independence
The sampled observations must be independent of one another.
Condition What to check
Randomization Data must come from a random sample (SRS), random assignment
condition in an experiment, or an unbiased sampling method.
10% condition If sampling without replacement, the sample size 𝑛 must be no more
than 10% of the population: 𝑛 ≤ 0.10 ⋅ 𝑁. This ensures approximate
independence.
Assumption 2 — Large Sample Size
The sample must be large enough for the Normal approximation to hold.
Condition What to check
Success/Failure Both 𝑛𝑝 ≥ 10 and 𝑛𝑞 ≥ 10 (using the population proportion 𝑝
condition when it is known; using 𝑝̂ when constructing a CI).
When the population proportion is known (e.g. testing against a claimed value),
use 𝑛𝑝 and 𝑛𝑞. When constructing a CI (population proportion unknown), use
𝑛𝑝̂ and 𝑛𝑞<.
Condition Check Summary Table
Condition Formula / Check Purpose
-------------------------------------------------------------------
Randomization Random sample / assignment? Independence
10% condition n ≤ 0.10 N ? Independence (w/o
replacement)
Success/Failure np ≥ 10 AND nq ≥ 10 Normality of
sampling dist.
9
STATISTICS II
SUMMARY WITH TI84 GUIDE
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 ...................................................... 6
Overview (Neutral) ...................................................................................................... 6
Course Structure (Less important) ............................................................................... 6
Course Content Topics (Neutral) ................................................................................. 7
What You Need From Statistics I (Important) ................................................................ 7
Chapter 10 — Sampling Distributions and Confidence Intervals for Proportions ................ 8
10.1 The Sampling Distribution of a Sample Proportion (Important) ............................... 8
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) ........................................................................... 10
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) ............................................. 12
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 ............................................................... 15
11.1 From Proportions to Means (Neutral) .................................................................. 15
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) ................................................................... 17
11.5 One-Sample t Confidence Interval (Important) .................................................... 18
11.6 Assumptions and Conditions (Important) ........................................................... 19
11.7 Interpretation of a Confidence Interval (Important) ............................................. 20
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
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)......................................... 27
12.5 Decision Rule: Significance Level 𝛼 (Important) .................................................. 27
12.6 Conclusion: In Words, In Context (Important) ..................................................... 28
12.7 Connection Between Confidence Intervals and Two-Sided Tests (Neutral) ........... 28
12.8 Critical Value Approach vs p-Value Approach (Neutral) ....................................... 29
12.9 Common Pitfalls and Misinterpretations (Important) ........................................... 30
12.10 Full Worked Example: Home Advantage in Baseball (Neutral) ............................ 30
Summary Table: One-Proportion z-Test ..................................................................... 31
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) .................................................................... 37
13.9 Effect Size and Statistical vs. Practical Significance (Important) .......................... 38
13.10 Power = 1 − 𝛽 (Important) ................................................................................ 38
13.11 Common Pitfalls (Important) ............................................................................ 39
13.12 Formula Reference Table (Important) ............................................................... 40
Chapter 14 — Comparing Two Means ............................................................................ 40
14.1 Two Independent Samples: Comparing 𝜇1 − 𝜇2 (Important) ................................ 40
14.2 The Two-Sample 𝑡-Test (Welch / Unpooled) (Important) ...................................... 41
14.3 Two-Sample 𝑡 Confidence Interval for 𝜇1 − 𝜇2 (Important) .................................. 42
14.4 The Pooled 𝑡-Test (Neutral) ................................................................................ 43
14.5 Conditions and Assumptions (Important) ........................................................... 44
2
, 14.6 Two-Proportion 𝑧-Test and CI: Comparing 𝑝1 − 𝑝2 (Important) ............................ 44
14.7 Paired vs. Independent Two-Sample: When Each Applies (Important) .................. 45
14.8 Common Pitfalls (Important).............................................................................. 46
14.9 Formula Summary Table .................................................................................... 47
14.11 Decision Guide — Which Test to Use? .............................................................. 47
Chapter 15 — Inference for Counts: Chi-Square Tests ................................................... 47
15.1 When to Use Chi-Square Tests (Important) ......................................................... 47
15.2 The Three Chi-Square Tests (Important) .............................................................. 48
15.3 The Chi-Square Test Statistic (Important) ........................................................... 49
15.4 Computing Expected Counts (Important) ........................................................... 50
15.5 Conditions for Validity (Important)...................................................................... 51
15.6 Standardized Residuals — Which Cells Drive Significance (Important) ................. 51
15.7 Relationship between Chi-Square (2 × 2) and the Two-Proportion z-Test (Important)
................................................................................................................................ 52
15.8 Running and Reading the Chi-Square Test on the TI-84 (Important) ...................... 52
15.9 Step-by-Step Procedure for Any Chi-Square Test (Important)............................... 53
15.10 Common Pitfalls (Important) ............................................................................ 53
15.11 Summary Table (Important).............................................................................. 54
Chapter 16 — Linear Regression and Inference for Regression ....................................... 54
16.1 Recap: Correlation (Chapter 4) (Important) ......................................................... 54
16.2 The Importance of Making Graphs — Anscombe’s Quartet (Important) ................ 55
16.3 The Linear Regression Model (Important) ............................................................ 56
16.4 Interpreting Slope and Intercept (Important) ....................................................... 57
16.5 Understanding Regression from Correlation (Neutral) ......................................... 57
16.6 Coefficient of Determination 𝑅2 (Important) ....................................................... 57
16.7 Standard Deviation of Residuals 𝑠𝑒 (Important)................................................... 58
16.8 Assumptions and Conditions (LINE) (Important) ................................................. 58
16.9 Outliers, Leverage, and Influential Points (Important) .......................................... 59
16.10 Extrapolation Warning (Neutral) ....................................................................... 60
16.11 Inference for Regression (Chapter 16) (Important) ............................................. 60
16.12 Inference for Predicted Values (Important)........................................................ 61
16.13 Reading TI-84 LinRegTTest Output (Important) ............................................... 62
16.14 Summary Table: Inference in Simple Linear Regression (Important) ................... 63
3
, 16.16 Common Pitfalls and What Can Go Wrong (Important) ...................................... 64
16.17 Quick Reference: Key Formulas (Important) ...................................................... 64
Chapter 17 — Inference for Regression and Residual Analysis ........................................ 65
17.1 The Population Regression Model (Important) ..................................................... 65
17.2 Sampling Distribution of the Slope 𝑏1 (Important) ............................................... 66
17.3 Hypothesis Test for the Slope (Important) ........................................................... 66
17.4 Confidence Interval for the Slope (Important) ..................................................... 67
17.5 Reading TI-84 LinRegTTest Output (Important) ................................................. 67
17.6 Standard Errors for Predicted Values (Important) ................................................ 68
17.7 Conditions for Inference — The LINE Assumptions (Important) ............................ 69
17.8 Unusual and Extraordinary Observations (Important) .......................................... 71
17.9 Extrapolation (Neutral) ...................................................................................... 72
17.10 Working with Summarized Data (Neutral) ......................................................... 72
17.11 Data Transformations (Important) .................................................................... 72
17.12 Overview of Inference for Regression ................................................................ 73
Chapter 23 — Non-Parametric Methods ........................................................................ 73
23.1 Why Non-Parametric Methods? (Important)........................................................ 73
23.2 Wilcoxon Rank-Sum / Mann–Whitney Test (Important) ........................................ 74
23.3 Kruskal-Wallis Test (Important) .......................................................................... 76
23.4 Wilcoxon Signed-Rank Test (Important) .............................................................. 76
23.5 Kendall’s Tau — Measuring Monotonicity (Neutral) ............................................. 78
23.6 Spearman’s Rank Correlation 𝜌 (Neutral) ........................................................... 78
23.7 Conditions Checklist Summary (Important) ........................................................ 79
Course Wrap-Up & Method Selector ............................................................................. 80
W.1 The Decision Framework (Important) .................................................................. 80
W.2 Decision Tree ..................................................................................................... 80
W.3 Complete Method Cheat-Sheet (Important) ........................................................ 82
W.4 Conditions Checklist for Each Test (Important) .................................................... 84
W.5 The LEGO Example — Applying the Selector (Important) ...................................... 84
W.6 Common Exam Pitfalls (Important) ..................................................................... 85
W.7 Full Exam Formula Reference (Non-Parametric) (Important) ................................ 86
4
,Statistics II — Course Summary
Course: Statistics for Business and Economics II (1009724BNR) — Vrije Universiteit
Brussel (VUB) Handbook: Business Statistics, 4th global edition — Sharpe, De Veaux &
Velleman
How to use this summary
This is the theory-only companion to Statistics II — Worked Solutions Manual. It distils all
course content (HOC1–HOC11 + LEGO recap docs) into roughly 55 pages, structured by
chapter. Worked exercises live in the separate Solutions Manual; pair the two documents
while studying.
Every subsection is tagged with a priority label so you can revise efficiently:
Tag Meaning What to do
(Important) Core exam material — high probability Study deeply, memorise
of appearing formulas
(Neutral) Standard course content — supports Read once carefully;
the Important parts understand the logic
(Less Side topics, advanced caveats, course Skim for awareness
important) logistics
The exam is closed-book multiple-choice (20 questions, 50 % theory + 50 %
exercises) with a higher-pass-mark scheme. The formula sheet and tables from
Canvas are provided.
Notation conventions
Symbol Meaning
𝜇, 𝜎, 𝑝 Population parameters (unknown)
𝑦‾, 𝑠, 𝑝̂ Sample statistics (estimates)
𝑛 Sample size
∗ Critical value from the standard Normal distribution
𝑧
∗
𝑡"# Critical value from the 𝑡-distribution with 𝑑𝑓 degrees of freedom
𝑆𝐸 Standard Error of a statistic
𝐻$ , 𝐻% Null hypothesis, alternative hypothesis
𝛼 Significance level
𝑝-value Probability of seeing data at least as extreme as observed under 𝐻$
𝛽$ , 𝛽& Population regression intercept and slope
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,𝑏$ , 𝑏& Sample regression intercept and slope
𝑦< Predicted (fitted) response from a regression model
𝑒 Residual (𝑦 − 𝑦<)
Common critical values (memorise!)
Confidence level Two-sided 𝑧 ∗ Right-tail 𝑧 ∗ (𝛼)
90 % 1.645 𝑧 ∗ = 1.282 for 𝛼 = 0.10
95 % 1.96 𝑧 ∗ = 1.645 for 𝛼 = 0.05
98 % 2.33 𝑧 ∗ = 2.054 for 𝛼 = 0.02
99 % 2.576 𝑧 ∗ = 2.326 for 𝛼 = 0.01
For 𝑡, the critical value depends on 𝑑𝑓 — read from the 𝑡-table (provided at the exam).
Chapter 0 — Course Introduction & Statistics I Recap
Overview (Neutral)
Statistics for Business and Economics II (course code 1009724BNR) is the follow-up to
Statistics I, using the handbook Business Statistics, 4th global edition, by Sharpe, De
Veaux & Velleman (3rd edition is also usable — a comparison table is on Canvas). The
course runs from February 2026. The lecturer is Evy Rombaut; seminars are led by Dennis
Verbist.
Course Structure (Less important)
Component Details
Lectures Monday 1–4 pm; theory + LEGO examples
Seminars Tuesday or Friday; exercises by hand and on the TI-84
Statistics Café Wednesday 2–4 pm; optional Q&A/drop-in
Evaluation:
• 10 % — intermediate test (on-campus, ~May 11; automatically transferred to 2nd
session — no retake)
• 90 % — multiple-choice exam: 50 % theory + 50 % exercises
– Must score ≥ 10/20 on the exam to pass the course
– Retake in 2nd session is possible if total score < 10/20
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,TI-84 on the exam: You must be able to (1) enter data into lists (or a matrix for chi-square),
(2) run the correct test/command from STAT ▸ TESTS or the DISTR menu, and (3) interpret
the output.
Course Content Topics (Neutral)
Module Handbook chapters
Sampling Chapter 10
Hypothesis testing Chapters 11–15
Correlation & Regression Chapters 4, 16–17
Non-parametric Methods Chapter 23
What You Need From Statistics I (Important)
The following concepts are assumed known. They are not re-taught but are used
constantly throughout the course.
Parameters vs. Statistics (Important)
• Parameter — a numerical summary of the population (fixed but usually unknown);
denoted with Greek letters, e.g. 𝜇, 𝜎, 𝑝.
• Statistic — a numerical summary computed from a sample; denoted with Roman
letters or hats, e.g. 𝑥‾, 𝑠, 𝑝̂ .
Population vs. Sample (Important)
• Population — the entire group you want to draw conclusions about.
• Sample — a subset of the population, studied because surveying the whole
population is infeasible.
Descriptive vs. Inferential Statistics (Important)
Type Goal
Descriptive Summarise and describe the data you have (histograms, means, proportions)
Inferential Draw conclusions about the population based on a sample
Statistics II is almost entirely about inferential statistics.
Types of Variables (Important)
• Categorical (qualitative) — values are labels/categories; summarised by
proportions/counts.
– Binary/dichotomous: only two possible values (e.g. success/failure, yes/no).
• Quantitative (numerical) — values are numbers; summarised by means, standard
deviations.
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, – Discrete: countable integer values.
– Continuous: any value on an interval.
The distinction matters because Chapter 10 deals exclusively with categorical variables
and proportions, while later chapters deal with means.
Key Distributions to Know (Important)
• Normal distribution 𝑁(𝜇, 𝜎) — bell-shaped, symmetric; characterised by the 68–95–
99.7 rule.
• Standard Normal 𝑁(0,1) — used for z-scores.
'()
• z-score: 𝑧 = *
Concept of Sampling Variability (Important)
Different samples yield different statistics. The sampling distribution is the distribution of
a sample statistic across all possible samples — understanding this is central to the entire
course.
Chapter 10 — Sampling Distributions and Confidence Intervals for
Proportions
10.1 The Sampling Distribution of a Sample Proportion (Important)
Setup
We have a population in which a proportion 𝑝 of individuals have some characteristic
(“success”). We draw a random sample of size 𝑛 and observe 𝑝̂ , the sample proportion:
number of successes in sample
𝑝̂ =
𝑛
𝑝 is the unknown parameter; 𝑝̂ is the statistic we observe.
The Sampling Distribution
Repeated sampling yields a sampling distribution of the proportion — bell-shaped and
centred at 𝑝 (verified by simulation). Under the right conditions:
𝑝𝑞
𝑝̂ ∼ 𝑁 I𝑝, K M, 𝑞 =1−𝑝
𝑛
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, Property Value
Centre (mean) 𝑝
Standard deviation 𝑝𝑞
𝑆𝐷(𝑝̂ ) = K
𝑛
Shape Approximately Normal (when conditions are met)
Key insight: even though we can only draw one sample in practice, we can use the known
sampling distribution model to make probability statements about where 𝑝̂ will fall relative
to 𝑝.
10.2 Assumptions and Conditions for the Normal Model (Important)
Two assumptions must hold, each checked via conditions:
Assumption 1 — Independence
The sampled observations must be independent of one another.
Condition What to check
Randomization Data must come from a random sample (SRS), random assignment
condition in an experiment, or an unbiased sampling method.
10% condition If sampling without replacement, the sample size 𝑛 must be no more
than 10% of the population: 𝑛 ≤ 0.10 ⋅ 𝑁. This ensures approximate
independence.
Assumption 2 — Large Sample Size
The sample must be large enough for the Normal approximation to hold.
Condition What to check
Success/Failure Both 𝑛𝑝 ≥ 10 and 𝑛𝑞 ≥ 10 (using the population proportion 𝑝
condition when it is known; using 𝑝̂ when constructing a CI).
When the population proportion is known (e.g. testing against a claimed value),
use 𝑛𝑝 and 𝑛𝑞. When constructing a CI (population proportion unknown), use
𝑛𝑝̂ and 𝑛𝑞<.
Condition Check Summary Table
Condition Formula / Check Purpose
-------------------------------------------------------------------
Randomization Random sample / assignment? Independence
10% condition n ≤ 0.10 N ? Independence (w/o
replacement)
Success/Failure np ≥ 10 AND nq ≥ 10 Normality of
sampling dist.
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