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Credit Sprint's Guide to AP® Statistics, Unit 5: Regression Analysis

AP® weight 10–20%

Unit 5 moves from one variable to two quantitative variables. You describe scatterplots, measure the strength of a linear relationship with the correlation coefficient, build a least-squares regression line and use residuals and r² to judge how well a linear model fits and what it can predict.

A scatterplot with a least-squares line explanatory variable xresponse y ŷ = a + bx residual = y − ŷ

What you need to know

  • A scatterplot shows the relationship between an explanatory variable (x) and a response variable (y). Describe the direction (positive or negative), form (linear or not), strength (how closely the points follow the pattern) and any unusual features, in context.
  • The correlation coefficient r measures the strength and direction of a linear relationship. It is always between −1 and 1, has no units, does not change if you change units or swap x and y, and only measures linear association.
  • A correlation near 0 means little linear relationship, not no relationship; a curved pattern can have r near 0. Correlation does not imply causation, because lurking variables can explain an association.
  • The least-squares regression line ŷ = a + bx minimizes the sum of squared residuals. The slope is b = r · (s_y ÷ s_x) and the intercept is a = ȳ − b·x̄. The line always passes through the point (x̄, ȳ).
  • Interpret the slope: for each 1-unit increase in x, the predicted y increases (or decreases) by about b units. Interpret the intercept: the predicted y when x is 0, which may not make sense if x = 0 is far from the data.
  • A residual is the actual value minus the predicted value: residual = y − ŷ. A positive residual means the point is above the line (the model under-predicted). The sum of residuals for a least-squares line is 0.
  • A residual plot graphs residuals against x or against predicted values. A random scatter with no pattern suggests a linear model is appropriate; a curved pattern suggests it is not.
  • The coefficient of determination r² is the proportion of the variation in y that is explained by the linear relationship with x. The standard deviation of the residuals s estimates the typical size of a prediction error.
  • Predicting within the range of the observed x-values is interpolation and is reasonable. Predicting outside that range is extrapolation, which is risky because the pattern may not continue.
  • In an experiment with random assignment, an association can be causal. In an observational study, a strong linear relationship does not show cause and effect.

Exam traps

Watch forThe slope of a least-squares line has the units y per x. r has no units. Do not confuse the two.
Watch forr² is not r. If r = −0.8, then r² = 0.64, meaning 64% of the variation in y is explained by the model.
Watch forCorrelation measures only linear relationships. A strong curved relationship can have r near 0.
Watch forDo not predict far outside the data (extrapolation) and do not assume cause from a regression on observational data.
Watch forResidual = actual − predicted (not the other way around), so a point above the line has a positive residual.

Key terms

  • Scatterplot: A graph that shows the relationship between two quantitative variables, one point per individual.
  • Association (scatterplot): Described by direction, form, strength and unusual features in context.
  • Positive association: As x increases, y tends to increase.
  • Negative association: As x increases, y tends to decrease.
  • Correlation r: A number from −1 to 1 that measures the strength and direction of a linear relationship.
  • r has no units: It does not change if you change units or switch x and y.
  • Lurking variable: A variable not included in the study that influences the relationship between the variables.
  • Regression line: A line used to predict y from x: ŷ = a + bx.
  • Least-squares regression line: The line that minimizes the sum of the squared residuals.
  • Slope b: The predicted change in y for each 1-unit increase in x; b = r · s_y ÷ s_x.
  • y-intercept a: The predicted y when x is 0; a = ȳ − b·x̄.
  • Predicted value ŷ: The value of y given by the regression line at a particular x.
  • Residual: Actual value minus predicted value: y − ŷ.
  • Residual plot: A scatterplot of residuals against x or against predicted values; no pattern suggests a good linear model.
  • Coefficient of determination r²: The proportion of the variation in y that is explained by the linear model.
  • Standard deviation of residuals s: The typical size of a prediction error in the units of y.

Practice questions

Five of the 80 Unit 5 questions in Credit Sprint. Every choice is explained in the app.

1. A scatterplot shows that as the number of hours of practice increases, free-throw percentage tends to increase, with points close to a straight line. Which description is best?
  1. No association
  2. A strong, positive, linear association
  3. A strong, positive, curved association
  4. A weak, negative, linear association
Show answer

A strong, positive, linear association. The direction is positive, the form is linear and the points are close to the line, indicating strength.

2. In a regression of y on x, the slope is 3.2. Which interpretation is correct?
  1. For each 1-unit increase in x, the predicted y increases by about 3.2 units
  2. When x is 0, y is 3.2
  3. For each 1-unit increase in y, x increases by 3.2 units
  4. The correlation between x and y is 3.2
Show answer

For each 1-unit increase in x, the predicted y increases by about 3.2 units. The slope is the predicted change in y for a one-unit change in x.

3. A residual plot of a regression shows a clear U-shaped curve. What does this suggest?
  1. The linear model fits well
  2. A linear model is not appropriate for these data
  3. The correlation is exactly 0
  4. The data contain no outliers
Show answer

A linear model is not appropriate for these data. A pattern in the residuals shows that the line misses a systematic feature such as curvature.

4. Why is extrapolation risky?
  1. The correlation becomes negative
  2. Regression lines have no predictions for large x
  3. The slope changes when x is large
  4. The relationship may not continue to be linear outside the range of the observed data
Show answer

The relationship may not continue to be linear outside the range of the observed data. The model was built from data in a limited range, and we have no evidence about other values.

5. Two variables have a correlation of r = 0.9 based on observational data. Which statement is the most appropriate?
  1. 90% of the variation is explained
  2. The relationship is not linear
  3. Changes in x cause changes in y
  4. The variables are strongly associated, but cause and effect cannot be concluded
Show answer

The variables are strongly associated, but cause and effect cannot be concluded. A strong correlation alone does not show that one variable causes the other.