AP Statistics, AP exams

Unit 5: Regression Analysis

Unit 5 covers the relationship between two quantitative variables. Scatterplots, correlation, the least-squares regression line, residuals, and the coefficient of determination are the content, and interpretation in context is the skill. In the revised framework this unit closes the course as a descriptive-analysis unit.

Position: 5 of 5 units
Topics covered: 6 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Statistics. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Scatterplots and association: Describing form, direction, strength, and unusual features of a bivariate relationship.
Correlation: What the correlation coefficient measures, its limitations, and why it does not establish causation.
Least-squares regression: Finding and interpreting the slope and intercept of the regression line in context.
Prediction, interpolation, and extrapolation: Using the line to predict, and recognizing when a prediction lies outside the data range.
Residuals and residual plots: Computing residuals and reading a residual plot to judge whether a linear model is appropriate.
Coefficient of determination: Interpreting the proportion of variability in the response explained by the model.

Study tips

How to practice Regression.

Section-specific guidance that follows from how this material is usually tested.

Interpret slope and intercept with units and context every time; a bare number earns no credit.
Read the residual plot before commenting on model fit, and say what pattern you see or do not see.
Rehearse the standard caution that association is not causation, phrased in the context of the given study.

Where this fits

Unit 5: Regression Analysis is 5 of 5 units in the AP Statistics guide. Use the full guide to balance it against everything else you have to review.

Back to the AP Statistics guide Official exam information

Sample Regression questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
A least-squares regression line predicting weekly sales in dollars from advertising spending in dollars is sales-hat = 250 + 3.2(advertising). What does the slope mean in context?
1. AEach additional dollar of advertising is associated with a predicted increase of $3.20 in weekly sales
2. BWeekly sales increase by 3.2 percent for each dollar of advertising
3. CWeekly sales are $3.20 when advertising spending is zero
4. DAdvertising causes weekly sales to rise by $3.20 per dollar
Show answer
Answer: A. Each additional dollar of advertising is associated with a predicted increase of $3.20 in weekly sales
Slope is the predicted change in the response for a one-unit increase in the explanatory variable, stated with units and as an association. The value 250 is the intercept, the prediction when advertising is zero, and observational regression data cannot support the causal wording in the last choice.
Key insight: Interpret a slope as a predicted change per one-unit increase, and say associated with rather than causes.
Question 2Foundational
A least-squares regression of exam score on hours studied has r^2 = 0.64. What does that value tell you?
1. AAbout 64% of the variability in exam score is accounted for by the linear model
2. BThe correlation between the variables is 0.64
3. CAbout 64% of the data points lie exactly on the regression line
4. DThe slope of the regression line is 0.64
Show answer
Answer: A. About 64% of the variability in exam score is accounted for by the linear model
The coefficient of determination is the proportion of the variability in the response explained by the least-squares model with that explanatory variable. Here the correlation is √0.64 = 0.8 in magnitude, not 0.64, and r^2 says nothing about how many points sit exactly on the line.
Key insight: r^2 reports explained variability in the response; r reports the strength and direction of the association.
Question 3Intermediate
A regression line is y-hat = 4 + 3x. For an observation with x = 6 and observed y = 19, what is the residual?
1. A-3
2. B3
3. C19
4. D22
Show answer
Answer: A. -3
The predicted value is 4 + 3(6) = 22, and the residual is observed minus predicted = 19 - 22 = -3. The negative sign means the model overpredicted this observation; reversing the subtraction produces the sign error 3.
Key insight: Residual equals observed minus predicted, so a negative residual means the model predicted too high.
Question 4Intermediate
A residual plot for a least-squares regression shows a clear U-shaped pattern. What does that indicate?
1. AA linear model is not appropriate for these data
2. BThe linear model fits the data well
3. CThe response variable has a large standard deviation
4. DThe correlation must be negative
Show answer
Answer: A. A linear model is not appropriate for these data
A residual plot should show random scatter about zero when a linear model is appropriate. A U-shaped pattern means the residuals are systematically positive, then negative, then positive again, which is the signature of curvature the line cannot capture.
Key insight: Read the residual plot for pattern, not size: curvature means the wrong model, not merely a poor fit.
Question 5Advanced
For a set of bivariate data the correlation is r = 0.6, the standard deviation of x is 4, and the standard deviation of y is 8. What is the slope of the least-squares regression line of y on x?
1. A1.2
2. B0.3
3. C2.4
4. D4.8
Show answer
Answer: A. 1.2
The slope is b = r times (s_y/s_x) = 0.6 times (8/4) = 1.2. Inverting the ratio of standard deviations gives 0.6 times (4/8) = 0.3, and multiplying r by s_x alone gives 2.4, which never uses the spread of the response.
Key insight: Slope scales the correlation by the ratio of the response spread to the explanatory spread.

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Unit 5: Regression Analysis