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Regression analysis practice problems AP Statistics, Unit 5

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 free-response question

Scatterplots, correlation, the least-squares regression line, residuals and r2, and the cautions of extrapolation and influential points. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Free-response practice
  5. Common mistakes
  6. Keep going

Key equations for regression analysis

ŷ = a + bxleast-squares regression line
b = r(sy/sx)slope
a = ȳ − b x̄intercept: the line passes through (x̄, ȳ)
residual = y − ŷactual minus predicted

The formulas and tables are provided in the exam. Answers within 2.5% are marked correct (5% for probabilities), so values from a table or a calculator both count. The full sheet is in our AP Statistics formula sheet.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
The regression line is ŷ = 3 + 2x. Predict y when x = 5.

Hint

Substitute x = 5.

Worked solution
  1. ŷ = 3 + 2 × 5 = 13.
Q2Warm-up
For that line, the actual value at x = 5 is 15. Find the residual.

Hint

Residual = actual − predicted.

Worked solution
  1. Residual = 15 − 13 = 2. The point lies above the line.
Q3Warm-up
A regression has r2 = 0.81. What does this mean?

Hint

r2 is the coefficient of determination.

Worked solution
  1. r2 measures the share of the variation in the response explained by the regression on x.
  2. The answer is C. Here r = ±0.9, not 0.81.

Exam-style questions

Questions at the level of the AP Statistics exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
For two variables, r = 0.8, sx = 4 and sy = 10. Find the slope of the least-squares regression line.

Hint

b = r(sy/sx).

Worked solution
  1. b = 0.8 × 10/4.
  2. b = 2.
Q5Exam-style
The same data have x̄ = 20 and ȳ = 50. Find the intercept of the regression line.

Hint

The line passes through (x̄, ȳ).

Worked solution
  1. a = ȳ − b x̄ = 50 − 2 × 20.
  2. a = 10.
Q6Exam-style
The correlation between two variables is −0.6. What fraction of the variation in y is explained by the regression on x?

Hint

Square r.

Worked solution
  1. r2 = (−0.6)2 = 0.36, so 36%.
Q7Exam-style
A regression of height on age uses children aged 2 to 10. Why is it unwise to use it to predict the height of a 30-year-old?

Hint

Is 30 inside the range of ages used to build the line?

Worked solution
  1. The line was fitted for ages 2 to 10; growth stops in adulthood, so the pattern does not continue.
  2. Predicting at age 30 is extrapolation: B.
Q8Challenge
Find the correlation r for the five points (1, 2), (2, 4), (3, 5), (4, 4), (5, 5).

Hint

Use your calculator's linear regression, or r = Σ(zxzy)/(n − 1).

Worked solution
  1. x̄ = 3, ȳ = 4. Σ(x − x̄)(y − ȳ) = 6, Σ(x − x̄)2 = 10, Σ(y − ȳ)2 = 6.
  2. r = 6/√(10 × 6) = 6/7.746.
  3. r = 0.775.

Free-response practice

One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.

Regression style · 10 points · about 22 minutes

A café records the daily high temperature x (°C) and the number of iced drinks sold y on 20 randomly chosen days. Computer output gives the least-squares line ŷ = −12.5 + 4.2x, with r = 0.91 and s = 6.3.

  1. (a) Interpret the slope in context.
  2. (b) Predict sales on a day with a high of 25°C, and find the residual if 96 iced drinks were sold.
  3. (c) Calculate and interpret r2.
  4. (d) Interpret s = 6.3 in context.
  5. (e) The café manager says hot weather causes more iced drink sales. Can the data support this? Explain.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

For each additional 1°C in daily high temperature, the predicted number of iced drinks sold increases by about 4.2.

Part (b)

ŷ = −12.5 + 4.2(25) = 92.5. Residual = 96 − 92.5 = 3.5.

Part (c)

r2 = 0.912 = 0.828. About 82.8% of the variation in daily iced drink sales is explained by the linear relationship with daily high temperature.

Part (d)

The actual number of iced drinks sold typically differs from the number predicted by the line by about 6.3 drinks.

Part (e)

Not from these data alone: this is an observational study, so other variables (such as weekends or school holidays) may be confounded with temperature.

Your mark: 0 of 10

Common mistakes in regression analysis

Keep going

Revise the ideas behind these questions:

Then try real exam questions: AP Statistics past papers, with official scoring guidelines. · All AP Statistics units