AP Statistics Unit 2 Question Wording Guide
If the question says...
Think...
What to do
“proportion/fraction/percent of variation in y
explained”
R-squared, R²
Use R² = r²; y must be the response variable.
“variation in y cannot be explained”
Unexplained variation
Use 1 - R² = 1 - r².
“for each 1-unit increase in x, predicted y...”
Slope
State the predicted change in y, in y-units, per 1 x-unit.
“when x is 0”
y-intercept
The model’s predicted y at x = 0; check whether that makes
sense in context.
“observed” compared with “predicted”
Residual
observed y - predicted y. Positive = model underestimated.
“is a linear model appropriate?”
Residual plot / scatterplot
Look for random residuals around 0 and no pattern.
“below minimum / above maximum / outside range”
Extrapolation
Do not rely on the LSRL there.
“unusually large or small x-value”
High leverage
Could strongly affect slope/intercept.
“given that,” “among,” “of female students”
Conditional distribution
Divide by the total of the named group.
A. Denominator Rule for Conditional Distribution
1. “All students/patients” or no subgroup specified: use the grand total.
2. “Given Medicine A,” “among adults,” “of female students”: use the total for that named group.
3. “Medicine A and reduction”: use the matching cell as the numerator. Without a “given” condition, divide by the grand
total.
B. Scatterplots, variables, and correlation
Term
Definition
Wording that should trigger it
Explanatory variable x
The input/predictor used to explain or predict another
variable. It is normally horizontal and comes after “from” or
“based on.”
“predict y from x,” “weight based on length,” “sales
versus advertising dollars”
Response variable y
The outcome being predicted or explained; normally vertical.
“predict runs from hits,” “predict plant height from pH”
Correlation coefficient r
A unitless number from -1 to 1 measuring the direction and
strength of the linear relationship between two quantitative
variables. It does not prove cause and effect.
“correlation coefficient,” “value of the correlation,”
“strength of the linear relationship”
Units do not change r
Changing feet to yards, for example, does not change
correlation.
“If ... expressed in yards instead of feet, how would
correlation r change?”
C. Least-squares regression line
The lecture calls this the LSRL: y-hat = a + bx. It is the line that minimizes the sum of squared residuals. See Two Variable
Data (Part II) (1).pdf, slides 4-13.
Term
Definition
Wording that should trigger it
Least-squares regression
line (LSRL)
A linear model that predicts y from x and minimizes the
sum of squared residuals.
“least-squares regression line/model,” “linear regression
analysis,” “regression equation”
Predicted value, y-hat
The model’s estimate of y for a selected x-value;
substitute x into y-hat = a + bx.
“predicted,” “point estimate,” “according to this model,”
“expect to sell”
Term
Definition
Wording that should trigger it
Slope b
For every 1-unit increase in x, predicted y changes by b yunits on average.
“for each ... increase, predicted ...”
Intercept a
Predicted y when x = 0. Interpret only if x = 0 is meaningful
and relevant.
“with 0 units of x”
Slope, intercept, or prediction?
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•
•
•
“For each additional/every 1-unit increase in x” = slope.
“When x is 0” = intercept.
“At x = 1,250 hits” or another particular x-value = prediction; substitute the x-value into the equation.
x-value outside sample range = extrapolation, even if the line itself is good inside the range.
D. Residuals and residual plots
Term
Definition
Wording that should trigger it
Residual
observed y - predicted y-hat; the vertical distance from a
point to the regression line.
“observed and predicted,” “calculate the residual,”
“absolute value of the residual”
Positive residual
Actual y is above predicted y-hat. The model
underestimated.
“Underestimate, because the residual is positive”
Negative residual
Actual y is below predicted y-hat. The model
overestimated.
“Overestimate, because the residual is negative”
Largest absolute residual
Point farthest vertically from the line; horizontal distance
does not matter.
“absolute value of the residual [is] largest”
Residual plot
Plot of residuals against x. Use it to assess whether a linear
model fits and whether spread is stable.
“residual plot ... support appropriateness of a linear
model?”
Good linear fit
Residuals are randomly scattered around 0, with no clear
pattern and roughly constant spread.
“no pattern,” “randomly distributed,” “centered at 0”
Nonlinear residual
pattern
A curved/U-shaped pattern means the line misses
systematic structure, so the linear model is not
appropriate.
“U-shaped pattern,” “curved pattern,” “linear model ...
right fit?”
Nonconstant variability
Fan-shaped residual spread (changing spread as x rises)
makes a linear fit less reliable.
“small variability for low x and larger ... for high x”
Standard deviation of
residuals, s
Typical vertical prediction error: observed y are typically
about s y-units from the LSRL’s predictions.
“standard deviation of the residuals ... interpret”;
“typical deviation between observed and predicted”
E. R2 and regression output
Term
Definition
Wording that should trigger it
Coefficient of
determination, R²
Proportion of variation in the response variable y explained
by the linear relationship with explanatory variable x. R² =
r².
“proportion/fraction/percent of variation in y explained”;
“R-sq”
Explained variation
The part of y’s variation accounted for by the regression on
x. State it in context as a percentage.
“36 percent of the variation in monthly sales could be
explained by monthly advertising”
Unexplained variation
The remaining variation: 1 - R². It is not 1 - r.
“proportion of the variation in y that cannot be explained”
Finding r from R²
r = ±√ (R²); choose the sign from the slope/direction.
“R2 is 36 percent ... What was the value of correlation
coefficient?”
R² wording Examples
Exact wording idea
Translation
“proportion of variation in CG explained by homework”
R²
“fraction of variation in y explained by least-squares regression of y on x”
R² = r².
“variation in y cannot be explained”
1 - R².
“Interpret R² in context.”
“__% of variation in [response] is explained by its linear relationship with
[explanatory variable].”
Do not confuse these: r gives direction and strength; R² is nonnegative and gives the explained proportion of y-variation. R²
always describes variation in y, not x.
F. Unusual and influential points
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•
•
Term
Definition
Wording that should trigger it
Outlier
A point unusually far vertically from the pattern/line; it often
has a large residual. It is not automatically influential.
“large residual”; “outliers are always influential points”
High-leverage point
A point with an unusually large or small x-value, far from
mean x. It can pull slope/intercept.
“most likely a high-leverage point”; “unusually large or
small x-value”
Influential point
Removing it substantially changes slope, intercept,
correlation, or the regression line.
“if removed ... affected”; “influential with respect to
regression”
Sum of squared
residuals
Total squared prediction error. It can change if a point is
removed, but is not the definition of influence.
“Which [quantity] is likely not affected by removal?”
Outlier = vertically far -> large residual.
High leverage = horizontally far -> possible slope/intercept change.
Influential = removal substantially changes the model.