How much context do I need to write on Unit 2 FRQs?

Enough to name the actual variables and their units. A slope interpretation should mention the real explanatory and response variables, the direction of change, and units. Bare numbers or generic x and y usually score partial credit at best.

Do I report r or r^2 from regression output?

Read the question carefully. Output shows R-Sq, which is r^2. If the question asks for the correlation r, take the square root of r^2 and give it the same sign as the slope. If it asks how much variation is explained, use r^2 as a percent.

When can I say one variable causes another?

Only when the data come from a well-designed experiment with random assignment. For observational data, a strong correlation cannot prove causation because a lurking variable may explain the association. On FRQs, state this explicitly.

What is the fastest way to lose points on these questions?

Common losses come from forgetting units, using generic variable names, confusing r with r^2, reversing observed and predicted in a residual, and judging fit from r^2 instead of the residual plot. Contradicting yourself within one part can also cost the point.

AP-STATS-2-FRQ

U2 FRQ Practice

Master AP Statistics Unit 2 free-response questions on scatterplots, correlation, regression, and residuals with a step-by-step FRQ attack plan.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on U2 FRQ Practice, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start U2 FRQ Practice free

What this lesson covers

Free-response questions in Unit 2 reward students who can describe relationships, interpret numbers in context, and justify conclusions rather than just crunch calculations. Almost every point on these FRQs comes with a hidden phrase: "in context." A slope means nothing until you attach the actual variables and units.

This guide assumes you already know how to build scatterplots, compute

r

, fit a least-squares line, and read residual plots. Here we focus on the skill that earns points: translating those tools into clear, complete written answers under time pressure. You will learn the structure graders look for, the exact phrases that lock in credit, and the traps that quietly cost partial points.

How Unit 2 FRQs Are Structured

A typical Unit 2 free-response question gives you either a scatterplot, a data table, or computer regression output, then asks a chain of parts (a), (b), (c) that build on each other. Early parts test description and interpretation; later parts test prediction, residual reasoning, and cautions about extrapolation or causation.

Graders score each part against a rubric that awards "essentially correct," "partially correct," or "incorrect." You rarely get credit for a bare number. The rubric almost always requires context — the real variable names and units — plus a correct statistical statement.

Here is how the common tasks map to what you must write.

Task	What earns the point
Describe a scatterplot	Direction, form, strength, unusual points, all in context
Interpret slope	"Predicted change in $y$ per one-unit increase in $x$ ," with units
Interpret intercept	Predicted $y$ when $x=0$ , noting if unrealistic
Interpret $r$ or $r^2$	Strength/direction, or percent of variation explained
Use residual plot	Comment on whether a line is appropriate

Read every part fully before writing. Later parts often reveal what a grader expects earlier, and reusing a value you already computed saves time.

Reading Computer Regression Output

Many FRQs replace hand calculation with a printed regression table. You must locate the right numbers quickly. A standard output lists a Constant (intercept) row and a predictor row, each with a Coef column. The regression equation is

\hat{y} = a + bx

, where

a

is the constant coefficient and

b

is the predictor coefficient.

Two more values appear below: S and R-Sq. Here

S

is the standard deviation of the residuals, describing typical prediction error in the units of

y

. R-Sq is

r^2

, the proportion of variation in

y

explained by the linear model. To get the correlation

r

, take

\pm\sqrt{r^2}

and match the sign to the slope: a negative slope means a negative

r

.

A frequent mistake is reporting

r^2

when the question asks for

r

, or forgetting the sign. Another is naming the response and explanatory variables backward. Always write the fitted equation using the actual variable names, for example

\widehat{\text{weight}} = 3.2 + 0.45(\text{length})

, not generic

x

and

y

. When you interpret

S

, say "the actual

y

values are typically about

S

units away from the values predicted by the line."

Interpretation Phrases That Earn Points

The single most tested skill is interpreting the slope. The template is: "For each additional one [unit of

x

], the predicted [

y

variable] increases (or decreases) by [slope] [units of

y

]." The word predicted matters — the line estimates an average change, not a guaranteed one.

Interpreting

r^2

: "About [

r^2

as a percent] of the variation in [

y

variable] is explained by the linear relationship with [

x

variable]." Do not say

r^2

explains the variation in

x

, and do not describe

r^2

as a probability.

Interpreting a residual: residual

=

observed

-

predicted, so

\text{residual} = y - \hat{y}

. A positive residual means the model underpredicted; the actual value sits above the line. State it in context: "The model underpredicted this student's score by 4 points."

Two cautions score their own points. Extrapolation: predicting outside the range of observed

x

values is unreliable. Causation: a strong correlation from observational data does not prove that

x

causes

y

; a lurking variable may be responsible. When a question asks whether you can conclude cause and effect, the answer for observational data is no, and you must explain why.

Using Residual Plots to Justify a Model

When an FRQ shows a residual plot, it is testing whether you can decide if a linear model is appropriate. The key idea: a residual plot with no leftover pattern — points scattered randomly above and below zero — supports using a line. A curved pattern (like a U shape) means a line is the wrong model, even if

r

looks strong.

Be precise in your justification. Do not just say "the residual plot looks good." Write "the residual plot shows no clear pattern and random scatter around zero, so a linear model is appropriate." If there is a pattern, say "the residual plot shows a curved pattern, indicating that a linear model does not fit well and a nonlinear model may be better."

Also watch for changing spread. If residuals fan out as

x

increases, predictions are less reliable for large

x

, and the equal-variability condition is questionable. A common trap is confusing a high

r^2

with a good fit; a strong correlation can still come from data that curves. The residual plot, not

r^2

alone, is the deciding evidence. Always tie your conclusion back to the specific variables in the problem.

A Time-Efficient FRQ Attack Plan

Work each part in a fixed order so you never omit context. First, underline exactly what the part asks — describe, interpret, predict, or justify. Second, identify the explanatory and response variables. Third, write your answer using their real names and units.

For a prediction, substitute into the equation and show the arithmetic:

\hat{y} = a + b(x_0)

. Report the predicted value with units. If the requested

x

lies outside the data range, add a sentence noting that this is extrapolation and may be unreliable.

Manage your minutes. A four-part question does not need an essay; two clean sentences per part usually secure the point. Avoid hedging with contradictory statements — if you write both a correct and an incorrect interpretation, graders may penalize the contradiction.

Finally, define any symbol you introduce. If you write

\hat{y}

, state what

y

represents. Rubrics reward communication: a correct idea buried in vague wording can be scored partial rather than essentially correct. Neat, contextual, complete sentences are worth as much as correct numbers.

Key terms

Residual.: The difference between an observed value and the value predicted by the regression line, $\text{residual} = y - \hat{y}$ . Positive means the model underpredicted.
Slope.: The coefficient $b$ in $\hat{y} = a + bx$ ; the predicted change in the response variable for each one-unit increase in the explanatory variable.
Coefficient of determination ( $r^2$ ).: The proportion of variation in the response variable explained by the linear relationship with the explanatory variable.
Correlation ( $r$ ).: A measure of the strength and direction of a linear relationship, ranging from $-1$ to $1$ ; its sign matches the slope.
Extrapolation.: Using a regression model to predict beyond the range of the observed explanatory values, which is unreliable.
Residual plot.: A scatterplot of residuals against the explanatory variable or predicted values, used to judge whether a linear model fits.
Standard deviation of residuals ( $s$ ).: The typical size of a prediction error, in the units of the response variable; labeled S in regression output.
Lurking variable.: An outside variable, not measured, that may explain an observed association and prevent a cause-and-effect conclusion.

Worked example

A researcher records the age (in years) and resale price (in hundreds of dollars) of 15 used cars of one model. Regression output gives: Constant Coef

= 182.4

, Age Coef

= -11.6

, R-Sq

= 0.81

. (a) Write the least-squares equation. (b) Interpret the slope in context. (c) A car is 6 years old and sold for 105 (hundreds of dollars). Find and interpret its residual. (d) Can the researcher conclude that increasing age causes lower price?

Part (a): Use

\hat{y} = a + bx

with the response variable price and explanatory variable age. The equation is

\widehat{\text{price}} = 182.4 - 11.6(\text{age})

, where price is in hundreds of dollars and age is in years.

Part (b): The slope is

-11.6

. For each additional year of age, the predicted resale price decreases by 11.6 hundreds of dollars (about 1160 dollars). Note "predicted" and the units — both are needed for full credit.

Part (c): First predict for age

= 6

\hat{y} = 182.4 - 11.6(6) = 182.4 - 69.6 = 112.8

. The residual is observed minus predicted:

105 - 112.8 = -7.8

. Interpret it: the actual price was 7.8 hundreds of dollars (about 780 dollars) lower than the model predicted, so the line overpredicted this car's price.

Part (d): No. This is observational data, not an experiment, so a strong association does not establish causation. A lurking variable — such as mileage or condition, which tend to worsen with age — could be driving the lower prices. State this reasoning explicitly to earn the point.

Practice questions

A regression of daily ice cream sales (in dollars) on temperature (in degrees Fahrenheit) yields

r^2 = 0.64

and a positive slope. Which statement is the correct interpretation?

The correlation is $r = 0.64$
About 64% of the variation in ice cream sales is explained by its linear relationship with temperature
Temperature causes 64% of ice cream sales
For each 1-degree increase in temperature, sales rise 64%

Answer: About 64% of the variation in ice cream sales is explained by its linear relationship with temperature

r^2

is the proportion of variation in the response variable (sales) explained by the linear model, so 64% is correct. The value

0.64

r^2

, not

r

; here

r = \sqrt{0.64} = 0.8

r^2

never describes causation or a per-unit change, which rules out the other choices.

A study fits

\widehat{\text{score}} = 52 + 3.4(\text{hours})

predicting exam score from hours studied. A student studies 5 hours and scores 71. Compute the residual and explain what it says about the model's prediction for this student.

Answer: The residual is $+2$ ; the model underpredicted this student's score by 2 points.

Predict first:

\hat{y} = 52 + 3.4(5) = 52 + 17 = 69

. The residual is observed minus predicted:

71 - 69 = 2

. A positive residual means the observed score is above the line, so the model underpredicted the student's actual score by 2 points. Always show the prediction arithmetic and state the result in context.

A residual plot for a linear model of plant height versus days shows a clear U-shaped (curved) pattern, even though

r = 0.92

. What should you conclude about the appropriateness of the linear model, and why?

Answer: A linear model is not appropriate because the residual plot shows a curved pattern, indicating the relationship is nonlinear.

A high correlation does not guarantee a good linear fit. The residual plot is the deciding evidence: random scatter supports a line, but a clear curved pattern means systematic error remains, so the true relationship is nonlinear. Report this even when

r

is close to 1, because

r

alone cannot detect curvature.

FAQ

How much context do I need to write on Unit 2 FRQs?: Enough to name the actual variables and their units. A slope interpretation should mention the real explanatory and response variables, the direction of change, and units. Bare numbers or generic $x$ and $y$ usually score partial credit at best.
Do I report $r$ or $r^2$ from regression output?: Read the question carefully. Output shows R-Sq, which is $r^2$ . If the question asks for the correlation $r$ , take the square root of $r^2$ and give it the same sign as the slope. If it asks how much variation is explained, use $r^2$ as a percent.
When can I say one variable causes another?: Only when the data come from a well-designed experiment with random assignment. For observational data, a strong correlation cannot prove causation because a lurking variable may explain the association. On FRQs, state this explicitly.
What is the fastest way to lose points on these questions?: Common losses come from forgetting units, using generic variable names, confusing $r$ with $r^2$ , reversing observed and predicted in a residual, and judging fit from $r^2$ instead of the residual plot. Contradicting yourself within one part can also cost the point.

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