AP Statistics Least-Squares Regression and Residuals Test Solutions and Scoring (PDF)

AP Statistics math test solutions: Least-Squares Regression and Residuals – Maths PDF

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This regression residuals test answers for AP Statistics covers least-squares regression and residuals (Common Core AP STAT 2.6, AP STAT 2.7, AP STAT 2.8, AP STAT 2.9, HSS-ID.B.6, HSS-ID.C.7), with a free PDF to print.

Each solution comes with a scoring guide. Give credit for correct substitutions, correct residual signs, and interpretations that include the words predicted, per unit, and the context.

AP Statistics · Common Core State Standards: AP STAT 2.6, AP STAT 2.7, AP STAT 2.8, AP STAT 2.9, HSS-ID.B.6, HSS-ID.C.7

Regression residuals test answers: worked solutions

Question 1 – Regression line [3 pts]

Data with the regression line.

(a) $\hat{y}=49.1+3.05(7)=70.45$.

(b) For each additional hour studied, the predicted score increases by 3.05 points.

70.45; 3.05 points per hour; -1.40.

Scoring: 1 point per part.

Question 2 – Interpret r squared and s [2 pts]

r squared: 88% of the variation in tips is explained by the bill total. s: actual tips typically differ from predicted tips by about $1.20.

88% explained; typical prediction error $1.20.

Scoring: 1 point for each interpretation.

Question 3 – Residual plot [3 pts]

A residual plot.

(a) The residuals form a U shape.

(b) No, the relationship is curved.

U pattern; not linear; point below the line.

Scoring: 1 point per part.

Question 4 – Slope from summary [3 pts]

$b=0.8\times\frac{4}{400}=0.008$. $a=22-0.008(3000)=-2$.

$\hat{y}=-2+0.008x$.

Scoring: 1 point for the slope formula, 1 point for b, 1 point for a.

Question 5 – Extrapolation [2 pts]

$32+2.4(40)=128$ inches.

128 inches (over 10 feet) is absurd; age 40 is far outside the data, so the prediction is unreliable extrapolation.

Scoring: 1 point for 128, 1 point for the extrapolation comment.

Question 6 – Influential point [2 pts]

A point far from the others.

It has an extreme x value and lies far from the trend of the other points.

Removing it would change the slope substantially, so it is influential.

Scoring: 1 point for the extreme x value, 1 point for its effect on the line.

Question 7 – Interpret the intercept [3 pts]

(a) $-12+9.5(30)=273$ calories.

(b) Each extra minute adds about 9.5 calories to the predicted total.

273; 9.5 calories per minute; not meaningful.

Scoring: 1 point per part.

Question 8 – Compute r [2 pts]

$r=-\sqrt{0.49}=-0.7$.

r = -0.7: a moderately strong negative linear association.

Scoring: 1 point for -0.7, 1 point for the description.

Total score

Total

20 points

Key facts behind this regression residuals test answers

Key facts

LSRL: y-hat = a + bx, with b = r sy/sx and a = y-bar – b x-bar; it passes through (x-bar, y-bar).
Interpret the slope as the predicted change in y per unit of x, in context.
Residual = actual – predicted; a good linear model has a residual plot with no pattern.
r squared is the percent of variation in y explained by the linear model; s is the typical prediction error.
Avoid extrapolation and check for influential points.

How to use this regression residuals test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All AP Statistics chapters: AP Statistics math.

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