AP Statistics Unit 5: Regression Analysis

Describing the relationship between two quantitative variables with scatterplots and correlation, fitting and interpreting a least-squares regression line, using residuals and residual plots to judge a linear model, and interpreting $r^2$ and s. (Inference for the slope is not part of the course that starts in fall 2026.)

Topics: 5.1, 5.2, 5.3, 5.4, 5.5 · free response

Key formulas and facts

• Describe a scatterplot by direction, form, strength and unusual features, in context
• Correlation: $r=\dfrac{1}{n-1}\sum\left(\dfrac{x_i-\bar x}{s_x}\right)\left(\dfrac{y_i-\bar y}{s_y}\right)$, with $-1\le r\le1$ and no units
• Least-squares line $\hat y=a+bx$ with $b=r\dfrac{s_y}{s_x}$ and $a=\bar y-b\bar x$ (the line passes through $(\bar x,\bar y)$)
• Residual $=y-\hat y$ (actual minus predicted)
• $r^2$: the proportion of the variation in y accounted for by the linear model with x
• $s=\sqrt{\dfrac{\sum(y_i-\hat y_i)^2}{n-2}}$: the typical size of a residual

Worked examples

• Slope from summaries: $r=0.75$, $s_x=4$, $s_y=10$, $\bar x=20$, $\bar y=65$ give $b=0.75\cdot\frac{10}{4}=1.875$ and $a=65-1.875(20)=27.5$, so $\hat y=27.5+1.875x$.
• Prediction and residual: with $\hat y=27.5+1.875x$, at $x=24$ the prediction is $72.5$; an actual value of 70 has residual $70-72.5=-2.5$.
• From data: for x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 5, technology gives $\hat y=2.2+0.6x$, $r\approx0.775$ and $r^2=0.6$.
• Sign of r from $r^2$: if $r^2=0.49$ and the slope is negative, $r=-0.7$.
• Units: converting x from inches to centimeters divides the slope by 2.54 but leaves r unchanged.

Common traps

• Interpreting the slope without the word "predicted" (or "on average"): individual y-values vary around the line.
• Computing a residual as predicted minus actual: it is actual minus predicted.
• Using r to judge whether a linear model is appropriate: always look at the residual plot for curvature.
• Concluding causation from a strong correlation in observational data.
• Extrapolating far beyond the observed x-values.
• Forgetting that r has the same sign as the slope when taking a square root of $r^2$.
• Reading $r^2$ as the percent of predictions that are correct: it is the proportion of the variation in y accounted for by the line.

Practice questions

40 multiple-choice questions (4 options each, as on the new exam) and 2 free-response questions worth 10 points each. A calculator is allowed throughout. Try each one before opening the answer. Each line of a scoring guide is one point, scored 0 or 1 and earned independently of the other lines, the style of the new course's 10-point free-response scoring (no E/P/I part scores). A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Topic 5.1: Scatterplots

8 multiple-choice questions.

Question 1 · Multiple choice

The scatterplot shows the hours 8 students studied and their quiz scores. Which describes the direction of the association?

1. There is no association.
2. Negative: students who studied more tended to score lower.
3. Positive: students who studied more tended to score higher.
4. Positive, because all the scores are above 60.

Question 2 · Multiple choice

Using the same scatterplot of hours studied and quiz score, how strong is the linear association?

1. Moderate
2. Strong
3. Weak
4. There is no linear association

Question 3 · Multiple choice

The scatterplot shows weekly screen time and average nightly sleep for 8 teens. Which describes the association?

1. Positive and strong
2. Negative and moderate
3. Negative and very strong (almost a perfect line)
4. No association

Question 4 · Multiple choice

A study looks at whether the number of hours a store is open each week helps predict its weekly sales. Which variable is the explanatory variable?

1. Weekly sales
2. The store's location
3. The number of hours the store is open each week
4. Both variables are response variables

Question 5 · Multiple choice

A scatterplot of the speed of a car (mph) and its fuel efficiency (mpg) rises to a peak near 50 mph and then falls. What is the best description of its form?

1. Curved (nonlinear)
2. No association, because r is near 0
3. Linear and positive
4. Linear and negative

Question 6 · Multiple choice

Which point in the scatterplot is an outlier in the regression sense (far from the pattern of the other points)?

1. (5, 13.0)
2. (1, 3.1)
3. (8, 9.8)
4. (4, 5.8)

Question 7 · Multiple choice

A scatterplot of the number of firefighters at a fire and the damage in dollars shows a strong positive association. What is the best explanation?

1. Sending more firefighters causes more damage.
2. The association must be due to chance.
3. A lurking variable, the size of the fire, affects both the number of firefighters and the damage.
4. Damage causes more firefighters to be born.

Question 8 · Multiple choice

The scatterplot shows the ages and prices of 8 used cars of the same model. Which describes the association?

1. Weak and negative
2. Moderate and positive
3. Strong, negative and linear
4. Strong, positive and linear

Topic 5.2: Correlation

8 multiple-choice questions.

Question 9 · Multiple choice

Calculate the correlation r for the hours-studied data: hours 2, 3, 4, 5, 6, 7, 8, 9 and scores 61, 64, 70, 69, 76, 80, 79, 86.

1. -0.978
2. 0.500
3. 0.956
4. 0.978

Question 10 · Multiple choice

A least-squares regression of y on x has $r^2=0.64$ and a negative slope. What is the correlation r?

1. -0.80
2. -0.64
3. -0.41
4. 0.80

Question 11 · Multiple choice

The correlation between height (inches) and weight (pounds) for a group of adults is 0.72. What is the correlation if heights are converted to centimeters and weights to kilograms?

1. 0.72 × 2.54
2. 0.72
3. It cannot be found without the data.
4. 0.72 ÷ 2.2

Question 12 · Multiple choice

Which statement about the correlation coefficient r is correct?

1. r measures the strength and direction of a linear association only.
2. r = 0 means there is no association of any kind.
3. r = 0.9 means 90% of the points are on the line.
4. A strong correlation shows that x causes y.

Question 13 · Multiple choice

Calculate r for the car data: ages 1-8 years and prices 27.5, 24.0, 21.8, 19.0, 17.2, 14.1, 12.5, 10.0 (thousand dollars).

1. -0.997
2. -0.500
3. 0.995
4. 0.996

Question 14 · Multiple choice

Removing an outlier that lies far from the linear pattern of the other points will most likely do what to r?

1. Move r closer to 0
2. Change the sign of r
3. Move r farther from 0 (make the linear association stronger)
4. Leave r unchanged

Question 15 · Multiple choice

For the screen-time data (screen time 12, 18, 25, 31, 36, 44, 50, 57 hours; sleep 7.6, 7.9, 7.2, 7.8, 7.0, 7.5, 7.3, 6.9 hours), what is r?

1. -0.950
2. -0.621
3. -0.386
4. 0.621

Question 16 · Multiple choice

A city finds r = 0.85 between ice cream sales and swimming-pool injuries over 52 weeks. Which conclusion is appropriate?

1. Ice cream causes pool injuries.
2. There is no association because the variables are unrelated.
3. There is a strong positive association, likely because hot weather increases both.
4. 85% of injuries are caused by ice cream.

Topic 5.3: Linear regression models

8 multiple-choice questions.

Question 17 · Multiple choice

A regression of the price of a used car (in thousands of dollars) on its mileage (in thousands of miles) gives $\widehat{\text{price}}=24.6-0.118(\text{mileage})$. What price does the model predict for a car with 60 thousand miles?

1. 7.08
2. 17.52
3. 24.60
4. 31.68

Question 18 · Multiple choice

For $\widehat{\text{price}}=24.6-0.118(\text{mileage})$ (price in \$1000s, mileage in 1000s of miles), what is the correct interpretation of the slope?

1. For each additional 1,000 miles, the price decreases by exactly \$118.
2. A car with no miles costs \$24,600.
3. For each additional 1,000 miles, the predicted price decreases by about \$118.
4. For each \$1,000 increase in price, mileage decreases by 118 miles.

Question 19 · Multiple choice

Using $\widehat{\text{price}}=24.6-0.118(\text{mileage})$, by how much does the predicted price change when mileage increases by 25 thousand miles? (Answer in thousands of dollars.)

1. -21.65
2. -2.95
3. -0.12
4. 2.95

Question 20 · Multiple choice

The car model was fit to cars with 10 to 90 thousand miles. Why is predicting the price of a car with 250 thousand miles risky?

1. The residual would be zero.
2. Prices cannot be predicted from mileage.
3. It is extrapolation: the linear pattern may not continue far outside the observed mileages.
4. The slope would become positive.

Question 21 · Multiple choice

A regression of a plant's height (cm) on days since planting gives $\hat y=3.2+0.85x$. What height does it predict at 20 days?

1. 4.0
2. 17.0
3. 20.2
4. 64.0

Question 22 · Multiple choice

In $\hat y=3.2+0.85x$ (height in cm, x = days since planting), what does the y-intercept 3.2 mean, and is it meaningful?

1. The intercept is the correlation between the variables.
2. The plant's height can never be below 3.2 cm.
3. Each day the plant grows 3.2 cm.
4. The predicted height at planting (day 0) is 3.2 cm; this is meaningful only if day 0 is within or near the range of the data.

Question 23 · Multiple choice

A vendor recorded the noon temperature and cold drinks sold on 9 days (table). Using the least-squares line, how many drinks does the model predict on a 28°C day?

1. 322.0
2. 349.6
3. 377.2
4. 395.6

Question 24 · Multiple choice

Why is the predicted value written $\hat y$ rather than y?

1. It is always larger than y.
2. It is the residual.
3. It is the mean of the x-values.
4. It is an estimate of the response from the model, not an actual observed value.

Topic 5.4: Residuals and residual plots

8 multiple-choice questions.

Question 25 · Multiple choice

Using $\widehat{\text{price}}=24.6-0.118(\text{mileage})$, a car with 60 thousand miles sold for \$19,200 (19.2 thousand). What is its residual (in thousands of dollars)?

1. -1.68
2. 1.68
3. 5.40
4. 17.52

Question 26 · Multiple choice

For $\hat y=3.2+0.85x$, a plant at x = 30 days has a residual of −1.4 cm. What was its actual height?

1. 25.5
2. 27.3
3. 28.7
4. 30.1

Question 27 · Multiple choice

The residual plot for a least-squares line is shown. Is a linear model appropriate?

1. Yes: the residuals add to about 0.
2. No: the residuals form a curved (U-shaped) pattern, so a linear model is not appropriate.
3. No: some residuals are negative.
4. Yes: the residuals are small.

Question 28 · Multiple choice

The residual plot for a least-squares line is shown. What does it suggest?

1. The model is curved.
2. The correlation must be 0.
3. The residuals show a fan shape, so the spread increases with x.
4. The linear model is appropriate: the residuals are randomly scattered around 0 with no pattern.

Question 29 · Multiple choice

For the temperature and drinks data, what is the residual for the 35°C day (450 drinks sold), using the least-squares line?

1. -8.5
2. 0.0
3. 8.5
4. 128.6

Question 30 · Multiple choice

What does a negative residual mean?

1. The actual value is below the value predicted by the regression line.
2. The point lies above the regression line.
3. The prediction was too low.
4. The slope is negative.

Question 31 · Multiple choice

A residual plot shows residuals that are small for small x and become much larger in size for large x (a fan shape). What does this tell you?

1. The model is perfect.
2. r must be negative.
3. Predictions are less precise for large x; the variability of y around the line increases with x.
4. The model is curved.

Question 32 · Multiple choice

For the car data (ages 1-8; prices 27.5, 24.0, 21.8, 19.0, 17.2, 14.1, 12.5, 10.0 thousand dollars), what is the residual for the 4-year-old car?

1. -0.75
2. -0.48
3. 0.30
4. 19.00

Topic 5.5: Least-squares regression

8 multiple-choice questions.

Question 33 · Multiple choice

For a set of students, the correlation between hours of sleep (x) and reaction time in ms (y) is r = −0.6. The standard deviations are $s_x=1.2$ hours and $s_y=40$ ms. What is the slope of the least-squares line?

1. -33.33
2. -20.00
3. -0.02
4. 20.00

Question 34 · Multiple choice

Using the same summary (r = −0.6, $s_x=1.2$, $s_y=40$) with $\bar x=7$ hours and $\bar y=320$ ms, what is the y-intercept of the least-squares line?

1. 180
2. 300
3. 320
4. 460

Question 35 · Multiple choice

Using the temperature and drinks data (table), what is the slope of the least-squares regression line?

1. 0.109
2. 0.999
3. 8.265
4. 9.183

Question 36 · Multiple choice

Using the temperature and drinks data, what is $r^2$?

1. 0.9000
2. 0.9500
3. 0.9979
4. 0.9989

Question 37 · Multiple choice

A regression of drinks sold on temperature has $r^2=0.998$. Which interpretation is correct?

1. About 99.8% of the variability in drinks sold is accounted for by the least-squares line with temperature.
2. The slope is 0.998.
3. Temperature causes 99.8% of drink sales.
4. 99.8% of the predictions are exactly correct.

Question 38 · Multiple choice

For the hours-studied data (table), what is the standard deviation of the residuals, s?

1. 0.978
2. 1.680
3. 1.940
4. 8.430

Question 39 · Multiple choice

The hours-studied regression has s ≈ 1.94 points. What does s mean?

1. Scores vary by exactly 1.94 points.
2. Each extra hour of study raises the predicted score by 1.94 points.
3. Actual quiz scores typically differ from the scores predicted by the line by about 1.94 points.
4. The correlation is 1.94.

Question 40 · Multiple choice

For the hours-studied data, what is the y-intercept of the least-squares line?

1. 3.42
2. 54.33
3. 59.33
4. 73.12

Free response

Write a complete answer, in context, as you would type it on the exam. The scoring guides are ours, written in the new course's style: 10 points per question, each point scored 0 or 1 and earned independently (no E/P/I part scores). They are not College Board scoring guidelines.

Question 41 · Free response (10 points)

A piano teacher recorded, for a random sample of 20 of her students, the hours of practice in the week before a recital and the student's recital score (0 to 100). Computer output for the least-squares regression of score on practice hours is shown.

1. (a) Write the equation of the least-squares regression line, defining any variables. [1 point]
2. (b) Interpret the slope in context. [2 points]
3. (c) A student practiced 12 hours and scored 66. Find and interpret her residual. [3 points]
4. (d) Find the correlation between practice hours and score. [2 points]
5. (e) Interpret $r^2$ and s in context. [2 points]

Question 42 · Free response (10 points)

A coffee shop recorded the number of baristas working (x) and the number of drinks served in a 15-minute rush (y) on 8 randomly chosen mornings (table).

1. (a) Describe the association between baristas working and drinks served. [2 points]
2. (b) Find the least-squares regression line and the correlation. [3 points]
3. (c) Use the line to predict drinks served with 9 baristas, and explain whether a prediction for 25 baristas would be reliable. [2 points]
4. (d) On the morning with 7 baristas, 24 drinks were served. Find the residual for that morning and interpret it in context. [3 points]