Week 15 — Quiz (auto-graded) · Linear Regression & Inference

Course: Introduction to Statistics (MATH 11) · Silver Oak University (fictional sample) · Prof. Rivera
Objective tested: Objective 8 — fit and interpret a simple linear regression model, including inference for the slope (interpreting a regression line/slope, predicting ŷ, residuals, r², inference for the slope).
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 15.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-15-qti.xml; the reusable item-bank entries and the Canvas placement block are at the bottom of this file.

Blueprint

No trick questions; distractors target the Week 15 misconceptions named in the lecture outline. All arithmetic is pre-computed.

Questions, key, and feedback

Q1 (MC). A regression line predicting a student's exam score (y) from hours studied (x) is ŷ = 50 + 4x. The slope of 4 means —
- A. A student who studies 0 hours is predicted to score 4
- B. For each additional hour studied, the predicted exam score increases by about 4 points ✅
- C. The exam has 4 questions
- D. The correlation between hours and score is 4
Feedback: The slope is the change in the predicted y for each one-unit increase in x — here, +4 points of score per extra hour studied (units matter). (A confuses the slope with the intercept; D confuses it with r, which can't exceed 1.)

Q2 (MC). Using the same line ŷ = 50 + 4x, the intercept of 50 represents —
- A. The predicted exam score for a student who studied 0 hours ✅
- B. The increase in score per extra hour
- C. The number of students in the study
- D. The highest score anyone can earn
Feedback: The intercept is the predicted y when x = 0 — here, the predicted score at 0 hours of study. (B is the slope.)

Q3 (MC). A regression line is ŷ = 20 + 3x. Predict ŷ when x = 5.
- A. 23
- B. 28
- C. 35 ✅
- D. 100
Feedback: Plug in x = 5: ŷ = 20 + 3(5) = 20 + 15 = 35. Prediction is just arithmetic. (A adds only one 3; B uses the wrong product.)

Q4 (MC). On a scatterplot with its regression line, a data point lies above the line (its actual y is higher than the line predicts). Its residual is —
- A. Positive ✅
- B. Negative
- C. Zero
- D. Equal to the slope
Feedback: Residual = observed − predicted. A point above the line has observed > predicted, so the residual is positive. (Below the line → negative; on the line → zero.)

Q5 (MC). For the line ŷ = 20 + 3x, a data point has x = 5 and an observed y of 40. The residual is —
- A. −5
- B. +5 ✅
- C. 35
- D. 75
Feedback: Predicted ŷ = 20 + 3(5) = 35. Residual = observed − predicted = 40 − 35 = +5 (the point is 5 above the line). (A reverses the subtraction; C is the predicted value, not the residual.)

Q6 (Matching). Match each regression term to its meaning.
| Term | Correct meaning |
|---|---|
| Slope (b₁) | How much the predicted y changes for each one-unit increase in x |
| Intercept (b₀) | The predicted y when x = 0 |
| Residual | Observed y minus predicted y for a single point |
| r² | The share (percent) of the variation in y that the line explains |
Feedback: Slope = "per-one-x" change; intercept = ŷ at x = 0; residual = observed − predicted; r² = share of variation explained (the correlation squared). (The classic mix-up is r² with the slope — different jobs.)

Q7 (MC). Two variables have a correlation of r = 0.8. The coefficient of determination r² is —
- A. 0.40
- B. 0.64 ✅
- C. 0.80
- D. 1.60
Feedback: r² = r × r = 0.8 × 0.8 = 0.64, i.e., about 64% of the variation in y is explained by the line. (C just repeats r; D doubles it.)

Q8 (Multiple answer — select all that apply). Which of the following statements about r² are true?
- A. r² is always between 0 and 1 ✅
- B. r² is the same thing as the slope of the regression line
- C. r² is the proportion of the variation in y explained by the line ✅
- D. r² equals the correlation r, squared ✅
- E. r² has units, such as points or hours
Feedback: r² is a unitless number on 0 to 1 (A), equal to r squared (D), giving the share of variation in y explained (C). It is not the slope (B) and has no units (E). (B and E are the named traps.)

Q9 (True / False). A regression of exam score on hours studied reports a p-value of 0.03 for the slope, tested against α = 0.05. Statement: "Because 0.03 < 0.05, we reject H₀: slope = 0, so the slope is statistically significant (there is evidence of a linear relationship)."
- True ✅
- False
Feedback: True. For inference on the slope you compare the p-value to α: since 0.03 < 0.05, reject H₀ (slope = 0) → the slope is significant — real evidence of a linear relationship, not just noise. (If p ≥ α, you'd fail to reject.)

Q10 (MC). The line ŷ = 50 + 4x was fit using students who studied between 1 and 6 hours. A student uses it to predict the score for someone who studies 40 hours, getting ŷ = 50 + 4(40) = 210. The best response is —
- A. The prediction of 210 is reliable because the formula always works
- B. This is extrapolation — 40 hours is far outside the data (1–6 hours), so the prediction is untrustworthy (and 210 is impossible on a 100-point exam) ✅
- C. It proves that studying 40 hours causes a score of 210
- D. The slope must be wrong
Feedback: Predicting far outside the range of the data is extrapolation, where regression lies — and 210 on a 100-point exam is impossible. The line is only trustworthy within ~1–6 hours. (C also wrongly leaps to causation from observational data.)

Answer key (quick reference)

Quality gate (self-checked): each single-answer item (1–5, 7, 9, 10) has exactly one correct option; the multiple-answer item (Q8) lists the three true statements (A, C, D) and excludes the two named traps (B "is the slope," E "has units"). All arithmetic is pre-computed and verified: Q3 prediction 20 + 3·5 = 35 (distractor A = 23 from adding a single 3); Q5 residual 40 − 35 = +5 (distractor A = −5 reverses the subtraction); Q7 r² = 0.8² = 0.64. No item asserts a fact outside the Week 15 course definitions, and no item asks the student to fit a line from raw data — every line and value is supplied.

Item-bank entries (for variants + the midterm/final)

All ten items are tagged course=MATH11 · week=15 · objective=8 · topic=linear-regression-and-inference and deposited in Item Bank: Week 15 — Linear Regression & Inference. The final (Week 16) and the per-term variant updates draw fresh items from this bank. (Tags: q1 interpret-slope, q2 interpret-intercept, q3 predict-yhat, q4 residual-sign, q5 compute-residual, q6 regression-terms-match, q7 r-squared-from-r, q8 r-squared-properties, q9 inference-slope-pvalue, q10 extrapolation-causation.)

Canvas placement block

canvas_object   = Quizzes::Quiz
title           = "Week 15 Quiz — Linear Regression & Inference"
assignment_group = "Quizzes"
points_possible = 10
grading_type    = points
due_offset_days = 6        # 6 days after module start (Sun Dec 13)
published       = true
shuffle_answers = true
provenance      = "~ Prof. Rivera's edition · Fall 2026 · built with thecoursemaker.com"