How do I know whether an FRQ wants a slope test or a slope confidence interval?

Read the question stem. Words like "is there evidence," "test the claim," or "significant relationship" call for a t-test of H_0: = 0. Words like "estimate the slope," "how much," or "construct an interval" call for a confidence interval b ± t^* SE_b. Both use df = n - 2.

What degrees of freedom do I use for slope inference?

Always df = n - 2, where n is the number of data points (pairs). You subtract 2 because the regression line estimates two parameters, the slope and the intercept. This differs from a one-sample mean, which uses n - 1.

Do I need to check conditions if the FRQ already gives me a residual plot?

Yes — the residual plot is provided precisely so you can check conditions. Explicitly state what the plot shows: a random horizontal band supports linearity and equal variance. Also address random sampling, independence (10% condition), and the normality of residuals. Naming conditions without referencing the evidence loses credit.

Can I conclude that x causes y if the slope test is significant?

No, not from observational data. A significant slope only shows convincing evidence of a linear association. Cause-and-effect conclusions require data from a randomized experiment. On FRQs, explicitly note when data are observational to avoid overstating your conclusion.

AP-STATS-9-FRQ

U9 FRQ Practice and Procedure Selection

Master AP Statistics Unit 9 FRQs: pick the right inference procedure, run slope t-tests and intervals from regression output, and write full-credit responses.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on U9 FRQ Practice and Procedure Selection, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

By Unit 9, you know every inference procedure the AP Statistics exam can throw at you — proportions, means, chi-square, and now regression slope. The hard part is no longer computing a test statistic; it is reading a scenario, choosing the correct procedure, and writing a response that earns every point. This lesson trains that judgment.

We will practice attacking Unit 9 free-response questions built around slope inference (both the confidence interval and the

t

-test), and we will sharpen the broader procedure-selection skill from CED 9.6. You already learned the mechanics of the slope interval and the slope test in earlier lessons; here we focus on execution under exam conditions: identifying which of the six-plus procedures a prompt requires, stating conditions, and communicating conclusions in context.

A Decision Framework for Procedure Selection

The single most tested judgment in Unit 9 FRQs is choosing the right procedure. Read the prompt for two clues: the type of variable(s) and the question being asked (estimate versus test). Use this map.

Data situation	Procedure
One categorical variable, one sample	1-proportion $z$ interval/test
One quantitative variable, one sample	1-sample $t$ interval/test for a mean
Two independent groups, quantitative	2-sample $t$
Paired quantitative measurements	Paired $t$ (differences)
Two categorical groups, proportions	2-proportion $z$
One categorical variable vs. hypothesized distribution	Chi-square goodness of fit
Two categorical variables	Chi-square independence/homogeneity
Two quantitative variables, linear relationship	$t$ for slope of regression

When you see computer regression output, a scatterplot, or the words slope, predictor, and response, you are in slope-inference territory. If the question asks whether a linear relationship exists or whether the slope differs from zero, run a

t

-test for slope. If it asks to estimate the true slope, build a confidence interval for slope. A common mistake is confusing paired data with two-sample data — look for whether each observation in one group is naturally matched to one in the other.

Reading Regression Output and Building Slope Inference

AP FRQs almost always hand you computer output rather than raw sums. You must locate four numbers: the estimated slope

b

, its standard error

SE_b

, the sample size

n

, and either the

t

-statistic or the

p

-value.

A typical table looks like this.

Predictor	Coef	SE Coef	T	P
Constant	12.40	3.10	4.00	0.001
Distance	0.85	0.20	4.25	0.000

The slope row (Distance) gives

b = 0.85

and

SE_b = 0.20

. Degrees of freedom are

n - 2

. The test statistic is

t = \frac{b - 0}{SE_b} = \frac{0.85}{0.20} = 4.25

, matching the T column.

For a confidence interval, use

b \pm t^{*} \cdot SE_b

with

df = n - 2

. For example, with

n = 20

df = 18

, and a 95% level,

t^{*} \approx 2.101

, giving

0.85 \pm 2.101(0.20) = 0.85 \pm 0.420

, or

(0.430, 1.270)

.

The reported

p

-value in the output is two-sided for

H_a: \beta \neq 0

. If your alternative is one-sided, halve it (when the slope is in the hypothesized direction). Never grab the Constant row by mistake — graders penalize using the intercept's standard error for slope inference.

Conditions: The LINER Framework

Slope inference requires more conditions than a mean, and FRQs award points for stating and checking them. Remember LINER: Linear, Independent, Normal, Equal variance, Random.

Linear: the true relationship between

x

and

y

is linear — check the scatterplot for a straight-line pattern and the residual plot for no curved pattern. Independent: observations are independent, often satisfied by random sampling and the 10% condition. Normal: for each

x

, the responses vary normally around the line — check that a histogram or dotplot of residuals is roughly symmetric with no strong skew. Equal variance (homoscedasticity): the residual plot shows roughly constant vertical spread across all

x

values. Random: the data come from a random sample or randomized experiment.

On the exam, you rarely have the true model, so you verify these using the residual plot and a plot of residuals. A key misconception: students check conditions for the raw data instead of the residuals. The residual plot is your primary tool — a random horizontal band supports both linearity and equal variance, while a fan shape or curve signals a violation. Always state what you observe, not just the condition name; "the residual plot shows no leftover pattern and constant spread" earns credit, while merely writing "linear: yes" does not.

Writing Full-Credit FRQ Responses

Scoring on inference FRQs follows a predictable four-part structure. Hitting each part deliberately protects your score.

Step	What graders want
State	Name the procedure and define parameters, e.g. $\beta$ = true slope of the population regression line relating $y$ to $x$
Plan	State $H_0: \beta = 0$ and $H_a$ , or the interval and confidence level; check all conditions with evidence
Do	Show the test statistic or interval formula with numbers, then the value and $p$ -value or bounds
Conclude	Compare $p$ to $\alpha$ (or interpret interval); state a conclusion in context about the linear relationship

The conclusion must be in context and must not overstate. For a test, write "Because

p < \alpha

, we reject

H_0

; there is convincing evidence of a linear relationship between (predictor) and (response)." For an interval, write "We are 95% confident the true slope is between (low) and (high) units of

y

per one-unit increase in

x

." A frequent point-loss: interpreting the confidence level as a probability about one specific interval, or saying "prove" instead of "convincing evidence." If zero lies inside a slope interval, note that no linear relationship is ruled out.

Avoiding the Classic Unit 9 FRQ Traps

Several errors recur on regression FRQs. First, do not confuse correlation with slope: a significant slope indicates a linear relationship exists, not that

x

causes

y

— causation requires a randomized experiment. If the data are observational, explicitly say a cause-and-effect conclusion is not justified.

Second, watch degrees of freedom: slope inference uses

df = n - 2

, not

n - 1

. Using the wrong

df

changes

t^{*}

and can cost accuracy points.

Third, mind the direction of the alternative. If the question suggests a positive association, use

H_a: \beta > 0

and remember the software

p

-value is two-sided.

Fourth, interpret the slope and

r^2

correctly when asked. The slope is the predicted change in

y

per one-unit increase in

x

;

r^2

is the proportion of variation in

y

explained by the linear model — not the proportion of points on the line.

Finally, when a part asks you to compare procedures or choose one, justify your choice by naming the variable types. Writing "both variables are quantitative and we are examining their linear relationship, so a

t

-test for the slope is appropriate" earns the identification point that a bare procedure name does not.

Key terms

Population regression slope ( $\beta$ ).: The true slope of the line relating the mean response to the explanatory variable in the entire population; estimated by the sample slope $b$ .
Standard error of the slope ( $SE_b$ ).: A measure of how much the sample slope $b$ typically varies from sample to sample; read directly from the SE Coef column of regression output.
t-test for slope.: Inference procedure testing $H_0: \beta = 0$ against an alternative, using $t = b / SE_b$ with $df = n - 2$ , to decide whether a linear relationship exists.
Confidence interval for slope.: An interval $b \pm t^{*} SE_b$ estimating the true slope $\beta$ with a stated level of confidence, using $df = n - 2$ .
Residual plot.: A plot of residuals versus $x$ or predicted values, used to check linearity and equal-variance conditions; a random horizontal band supports the model.
LINER conditions.: The requirements for slope inference: Linear, Independent, Normal residuals, Equal variance, and Random data collection.
Procedure selection.: The skill of matching a scenario to the correct inference method based on the number and type of variables and whether an estimate or a test is requested.
Coefficient of determination ( $r^2$ ).: The proportion of variation in the response variable explained by the linear regression model.

Worked example

A researcher randomly sampled 25 used cars and recorded each car's age (years) and price (thousands of dollars). Software output for the regression of price on age gives: slope coefficient = -1.42, SE of slope = 0.35, intercept = 21.6. The residual plot shows a random horizontal band. At the 0.05 significance level, is there convincing evidence of a negative linear relationship between age and price?

State: Let

\beta

be the true slope of the population regression line predicting price from age. We test

H_0: \beta = 0

versus

H_a: \beta < 0

(negative relationship). We use a

t

-test for the slope.

Plan — conditions: Linear and Equal variance are supported because the residual plot shows a random horizontal band with no curve and constant spread. Random is met since the cars were randomly sampled. Independent is reasonable if 25 cars are less than 10% of all used cars. We assume the Normal condition for residuals holds. Conditions are met.

Do: Compute the test statistic with

df = n - 2 = 25 - 2 = 23

t = \frac{b - 0}{SE_b} = \frac{-1.42}{0.35} = -4.06

For a lower-tailed test with

df = 23

, this

t

gives a very small

p

-value, approximately

0.0002

.

Conclude: Because

p \approx 0.0002 < 0.05

, we reject

H_0

. There is convincing evidence of a negative linear relationship between a used car's age and its price — older cars tend to have lower prices. Because this is observational data, we cannot conclude that age causes the price change.

Practice questions

A study measures the height and arm span of 30 randomly selected students. Regression output gives slope = 0.88 with SE = 0.11. Which procedure and degrees of freedom should be used to build a 95% confidence interval for the true slope?

t-interval for slope with df = 30
t-interval for slope with df = 28
z-interval for slope with df = 29
2-sample t-interval with df = 29

Answer: t-interval for slope with df = 28

Both variables are quantitative and we want to estimate the linear relationship's slope, so we use a

t

-interval for slope. Degrees of freedom equal

n - 2 = 30 - 2 = 28

. Slope inference always uses the

t

distribution (not

z

) and subtracts 2 for the two estimated parameters (slope and intercept).

An FRQ gives regression output for predicting exam score from hours studied, with slope = 4.5, SE of slope = 2.6, and n = 12. Conduct a test at the 0.05 level to determine whether there is a linear relationship between hours studied and exam score. Show your hypotheses, test statistic, and conclusion.

Answer: Fail to reject H0; there is not convincing evidence of a linear relationship.

Let

\beta

be the true slope. Hypotheses:

H_0: \beta = 0

versus

H_a: \beta \neq 0

. With

df = 12 - 2 = 10

, the statistic is

t = 4.5 / 2.6 \approx 1.73

. For a two-sided test with

df = 10

, the

p

-value is roughly

0.11

, which is greater than

0.05

. We fail to reject

H_0

: there is not convincing evidence of a linear relationship between hours studied and exam score. Note that failing to reject does not prove no relationship exists — the sample may simply be too small to detect one.

A prompt describes measuring blood pressure of the same 40 patients before and after taking a medication and asks whether the medication changed blood pressure on average. Which inference procedure is appropriate, and why?

Answer: A paired t-test on the differences (before minus after).

Each patient provides two matched measurements, so the data are paired, not two independent samples. The correct procedure is a paired

t

-test using the differences for each patient. Students often wrongly pick a 2-sample

t

-test; the key clue is that the two measurements come from the same individual, creating natural pairs. Because the response is quantitative and we compare a mean difference to zero, this is a mean-difference test, not slope inference.

FAQ

How do I know whether an FRQ wants a slope test or a slope confidence interval?: Read the question stem. Words like "is there evidence," "test the claim," or "significant relationship" call for a $t$ -test of $H_0: \beta = 0$ . Words like "estimate the slope," "how much," or "construct an interval" call for a confidence interval $b \pm t^{*} SE_b$ . Both use $df = n - 2$ .
What degrees of freedom do I use for slope inference?: Always $df = n - 2$ , where $n$ is the number of data points (pairs). You subtract 2 because the regression line estimates two parameters, the slope and the intercept. This differs from a one-sample mean, which uses $n - 1$ .
Do I need to check conditions if the FRQ already gives me a residual plot?: Yes — the residual plot is provided precisely so you can check conditions. Explicitly state what the plot shows: a random horizontal band supports linearity and equal variance. Also address random sampling, independence (10% condition), and the normality of residuals. Naming conditions without referencing the evidence loses credit.
Can I conclude that x causes y if the slope test is significant?: No, not from observational data. A significant slope only shows convincing evidence of a linear association. Cause-and-effect conclusions require data from a randomized experiment. On FRQs, explicitly note when data are observational to avoid overstating your conclusion.

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