How do I know whether to use a one-sample or two-sample t-procedure?

Ask whether the data come from one group or two, and whether they are paired. A single sample or matched pairs (before/after, twins) uses one-sample t on the values or the differences. Two separate independent samples or two treatment groups use two-sample t. The key trap is paired data, which look like two columns but must be reduced to differences and analyzed as one sample.

What exactly must a conclusion include to get full credit?

Three parts: a comparison (p-value versus , or whether a value is captured in the interval), a decision (reject or fail to reject H_0), and context describing the real-world parameter. Missing context is the most common reason otherwise-correct conclusions lose a point.

Do I have to show the t-formula, or can I just use my calculator?

You may use calculator procedures, but you must name the test and report the inputs, test statistic, degrees of freedom, and p-value (or interval). Writing only a final number without identifying the procedure or showing the setup risks losing the 'Do' point. Naming the procedure and listing inputs makes your work verifiable.

How do I check the Normal condition when the sample is small?

If the population is stated to be normal, you are done. If not and n < 30, you must reference a graph of the sample data (dotplot, boxplot, or histogram) and state that it shows no strong skew or outliers. If n ≥ 30, cite the Central Limit Theorem. For two samples, check each group separately.

AP-STATS-7-FRQ

U7 FRQ Practice

Master Unit 7 AP Statistics FRQs on inference for means: pick the right t-procedure, check conditions, show work, and write conclusions that earn full credit.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on U7 FRQ Practice, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Unit 7 free-response questions reward more than a correct number — they reward a clearly structured argument. Readers score whether you identified the right procedure, verified conditions, computed carefully, and interpreted results in context. This lesson does not re-teach the t-procedures themselves; instead it shows you how to attack a means-inference FRQ under time pressure so nothing scoreable slips through the cracks.

You will learn a repeatable four-part template, how to decide between one-sample and two-sample tools, the exact language that earns condition and conclusion points, and the mistakes that quietly cost partial credit. Treat every FRQ like a checklist, and you convert scattered knowledge into reliable points.

Choosing the Right Procedure

The first scoreable decision on a means FRQ is naming the correct inference procedure. Read the prompt twice and ask three questions: Is the parameter a single mean or a difference of means? Am I estimating (interval) or testing a claim (test)? Are the data paired or independent?

Paired data are the classic trap. If each subject provides two measurements (before/after, left/right, twin pairs), you analyze the differences with a one-sample t-procedure — not a two-sample procedure. Independent groups (two separate random samples or two treatment groups) call for two-sample t.

Situation	Procedure
Estimate one population mean	One-sample $t$ -interval
Test a claim about one mean	One-sample $t$ -test
Before/after or matched pairs	One-sample $t$ on differences
Compare two independent groups (estimate)	Two-sample $t$ -interval
Compare two independent groups (test)	Two-sample $t$ -test

State the procedure by name at the top of your response. It costs one sentence and signals to the reader exactly what you intend to do. If a prompt asks whether results 'provide convincing evidence,' that phrasing means a hypothesis test; 'estimate' or 'how much' signals a confidence interval.

The Four-Part FRQ Template

Every means-inference FRQ can be answered with the same skeleton. Memorize it so you never omit a component.

1. State. Name the procedure and define parameters in context (e.g.,

\mu

= true mean commute time). For a test, write

H_0

and

H_a

using symbols and words.

2. Plan. Check conditions: Random (stated random sampling or random assignment), and Normal/large sample. The Normal condition is met if the population is stated normal,

n \geq 30

(CLT), or a graph of the data shows no strong skew or outliers. For two samples, check each group.

3. Do. Report the test statistic and p-value (test) or the interval (CI). Show the formula setup or name the calculator procedure with inputs. Include

df

when relevant.

4. Conclude. For a test: compare p-value to

\alpha

, then write a context sentence — 'Because

p < \alpha

, we reject

H_0

; there is convincing evidence that...'. For an interval: 'We are 95% confident the true mean (difference) is between ___ and ___,' then answer any follow-up question the interval addresses.

Skipping any part typically caps your score. Even if arithmetic goes wrong, correctly stating hypotheses and checking conditions earns partial credit.

Conditions and Conclusions That Earn Points

Readers use specific language cues. Vague statements lose points even when the idea is right.

For the Random condition, cite the prompt: 'The problem states a random sample of 40 households.' Do not just write 'random — check.' For random assignment in experiments, name it explicitly.

For the Normal/Large Sample condition, state which justification applies. If

n < 30

and no population information is given, you must reference a graph: 'The dotplot of the sample shows no strong skew or outliers, so the t-procedure is reasonable.' If

n \geq 30

, cite the Central Limit Theorem.

For conclusions, three elements must appear: a comparison (p-value vs.

\alpha

, or capturing a value in the interval), a decision (reject / fail to reject), and context (what the parameter means in the real scenario). A conclusion missing context is the single most common point loss on Unit 7 FRQs.

Avoid absolute language. Never say a test 'proves' something or that you 'accept

H_0

.' Write 'fail to reject

H_0

' and 'not enough evidence.' When failing to reject, do not conclude the null is true — conclude there is insufficient evidence against it. These distinctions are exactly what graders scan for.

Interpreting Intervals and Linking to Tests

FRQs love to combine an interval with a follow-up conclusion. If a two-sample interval for

\mu_1 - \mu_2

contains 0, you cannot conclude the means differ; if it lies entirely above 0, group 1's mean is plausibly larger. Say this explicitly and tie it to the question asked.

A correct interval interpretation describes the parameter, not the data: 'We are 95% confident that the true difference in mean battery life is between 1.2 and 4.8 hours.' Do not say '95% of samples' or '95% of batteries.' The confidence level describes the long-run success rate of the method, which is a separate interpretation the exam sometimes requests separately.

When a prompt shows a computed interval and asks whether a claimed value is plausible, check whether that value falls inside the interval. Inside means plausible; outside means the data provide evidence against it. This mirrors a two-sided test at the corresponding significance level, and stating that connection can strengthen your response.

Always answer the exact question. If asked 'Does the interval support the manager's claim?', end with a direct yes/no sentence grounded in whether the claimed value is captured.

Key terms

Paired data.: Two measurements from the same subject or matched pairs; analyzed by applying one-sample t-procedures to the differences.
Two-sample t-procedure.: Inference comparing means of two independent groups, using a t-statistic for $\mu_1 - \mu_2$ with software-computed degrees of freedom.
Normal/Large Sample condition.: Requirement that the sampling distribution of the mean is approximately normal, met via a stated normal population, $n \geq 30$ , or a graph free of strong skew and outliers.
p-value.: The probability of a test statistic as extreme as observed, assuming $H_0$ is true; compared to $\alpha$ to decide.
Confidence level.: The long-run proportion of intervals built by the same method that would capture the true parameter.
Fail to reject.: The correct outcome when evidence is insufficient; it does not confirm that $H_0$ is true.
Context sentence.: A concluding statement that references the actual real-world parameter, required for full conclusion credit.

Worked example

A trainer claims a new stretching routine reduces sprint times. Ten athletes are timed in the 100 m before and after four weeks of the routine. The mean difference (before minus after) is 0.18 seconds with standard deviation 0.22 seconds. A dotplot of the differences shows no strong skew or outliers. Is there convincing evidence at

\alpha = 0.05

that the routine reduces sprint times?

State: Because each athlete is measured twice, the data are paired. Use a one-sample t-test on the differences

d = \text{before} - \text{after}

. Let

\mu_d

= true mean reduction in sprint time.

H_0: \mu_d = 0

versus

H_a: \mu_d > 0

(a positive difference means faster after).

Plan: Random — assume the athletes are a representative/random sample (state this assumption). Normal —

n = 10 < 30

, but the dotplot of differences shows no strong skew or outliers, so the t-procedure is reasonable.

Do: The test statistic is

t = \frac{\bar{d} - 0}{s_d / \sqrt{n}} = \frac{0.18}{0.22 / \sqrt{10}} = \frac{0.18}{0.0696} \approx 2.59.

With

df = 9

, the one-sided p-value is approximately

0.015

.

Conclude: Because

p \approx 0.015 < 0.05 = \alpha

, we reject

H_0

. There is convincing evidence that the stretching routine reduces the true mean 100 m sprint time for these athletes.

Practice questions

Researchers compare fuel efficiency of two independent random samples of cars from two brands. Which procedure is appropriate to estimate how much the mean efficiencies differ?

One-sample t-interval for a mean
One-sample t-test on differences
Two-sample t-interval for a difference of means
Two-sample t-test for a difference of means

Answer: Two-sample t-interval for a difference of means

The samples are independent (not paired), and the word 'estimate how much they differ' signals an interval rather than a test. That combination points to a two-sample t-interval for

\mu_1 - \mu_2

. A test would be used only if asked whether a difference exists, and one-sample tools apply to a single mean or to paired differences.

A 95% confidence interval for the difference in mean scores

\mu_A - \mu_B

(-1.5, 4.2)

. Explain what this interval tells you about whether the two population means differ, and state your reasoning.

Answer: Because the interval contains 0, there is not convincing evidence that the two population means differ.

An interval for a difference that captures 0 means a difference of zero is plausible, so we cannot conclude the means are different at the corresponding significance level. A full-credit answer also interprets the interval — we are 95% confident the true difference

\mu_A - \mu_B

lies between -1.5 and 4.2 — and explicitly ties the presence of 0 to the lack of evidence for a difference.

On a one-sample t-test a student computes

t = 1.4

with

df = 24

and a two-sided p-value of

0.17

, then writes 'Since p > 0.05, we accept the null hypothesis that the mean equals 100.' Identify and correct the error.

Answer: The student should write 'fail to reject the null hypothesis,' not 'accept' it.

We never accept

H_0

. A large p-value means we lack sufficient evidence against the null, so we fail to reject it — but the true mean could still differ from 100. The corrected conclusion: 'Because

p = 0.17 > 0.05

, we fail to reject

H_0

; there is not convincing evidence that the true mean differs from 100.'

FAQ

How do I know whether to use a one-sample or two-sample t-procedure?: Ask whether the data come from one group or two, and whether they are paired. A single sample or matched pairs (before/after, twins) uses one-sample t on the values or the differences. Two separate independent samples or two treatment groups use two-sample t. The key trap is paired data, which look like two columns but must be reduced to differences and analyzed as one sample.
What exactly must a conclusion include to get full credit?: Three parts: a comparison (p-value versus $\alpha$ , or whether a value is captured in the interval), a decision (reject or fail to reject $H_0$ ), and context describing the real-world parameter. Missing context is the most common reason otherwise-correct conclusions lose a point.
Do I have to show the t-formula, or can I just use my calculator?: You may use calculator procedures, but you must name the test and report the inputs, test statistic, degrees of freedom, and p-value (or interval). Writing only a final number without identifying the procedure or showing the setup risks losing the 'Do' point. Naming the procedure and listing inputs makes your work verifiable.
How do I check the Normal condition when the sample is small?: If the population is stated to be normal, you are done. If not and $n < 30$ , you must reference a graph of the sample data (dotplot, boxplot, or histogram) and state that it shows no strong skew or outliers. If $n \geq 30$ , cite the Central Limit Theorem. For two samples, check each group separately.

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