- A full-credit inference answer has four parts: state the parameter and hypotheses, plan (name the procedure and check conditions), do the calculation, and conclude in context.
- Check each condition with the numbers and words of the question: random, 10% (when sampling without replacement) and normality (large counts or a large sample).
- Name the procedure in full, for example “one-sample z-test for a proportion”.
- Link the conclusion to the p-value or the interval, and never say you have “proved” or “accepted” the null hypothesis.
- Below: two model answers with every number checked, a sentence-starter template and five practice questions.
The problem: the maths was right, but the marks were not there
Amira is an AP Statistics student. In her mock exam she found a z-statistic of 2.26 and a p-value of 0.012. Both correct. Yet she lost much of the credit for that part.
Her hypotheses had no parameter. Her conditions said only “SRS, normal”. Her last line was “Reject H0, so p > 0.5.” The calculation was perfect. The communication was not.
This is the most common story in AP Statistics free response inference questions. The College Board’s course description stresses that students must identify the parameter and hypotheses, choose a procedure, check conditions, calculate, and give a conclusion in context linked to the p-value[1]. It also notes that students often lose credit because they check conditions superficially, without connecting them to the problem[1].
This is a writing skill, and a template fixes it. It also works for Cambridge 9709 S2, IB Maths and first-year university courses.
What a full-credit inference answer contains
Every inference question asks either “Is there convincing evidence that…?” (a significance test) or “Estimate…” (a confidence interval). Both use the same four steps, which many AP textbooks teach as State, Plan, Do, Conclude[2].
| Step | Significance test | Confidence interval |
|---|---|---|
| State | Define the parameter in words. Write H0 and Ha. Give α. | Define the parameter in words. Give the confidence level. |
| Plan | Name the test. Check random, 10% and normality with numbers. | Name the interval. Check the same three conditions. |
| Do | Formula with values, test statistic, df if relevant, p-value. | Formula with values, the interval. |
| Conclude | Compare p-value with α. Reject or fail to reject H0. Say what it means in context. | “We are 95% confident that the interval from … to … captures [parameter in context].” |
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Step 1, State: the parameter and the hypotheses
Say exactly what you are making a claim about: a parameter, which describes the population, not the sample.
- Weak: “p = proportion who said yes.” (Of whom? The sample or the population?)
- Strong: “p = the true proportion of all students at this school who support a later start time.”
Hypotheses must be stated in terms of population parameters, not sample statistics[1]. H0: p = 0.50 is right; H0: p̂ = 0.50 is wrong, because a statistic is already known. Then give α, usually 0.05. More help: null vs alternative hypothesis.
Step 2, Plan: name the procedure and check the conditions
“z-test” alone is vague. “One-sample z-test for a population proportion” is complete. Then check three conditions, each linked to the question.
| Condition | Why it matters | Proportions | Means |
|---|---|---|---|
| Random | Lets you generalise (random sample) or infer cause (random assignment). | “The data come from a random sample of 200 students.” | Same, for each sample or group. |
| 10% | Keeps observations close to independent when sampling without replacement. | 200 ≤ 0.10 × 2,400 = 240. | Same, for each sample. |
| Normality | Makes the sampling distribution approximately normal. | Test: np0 ≥ 10 and n(1 − p0) ≥ 10. Interval: observed successes and failures ≥ 10. | n ≥ 30, or a normal population, or a small sample whose graph shows no strong skew or outliers. |
The latest course description calls the third check the “normality condition” and uses counts of at least 10 for proportions and n ≥ 30 for means[1]. Many textbooks call the proportion version “Large Counts”. Either name is fine if you show the numbers. Background: the sampling distribution of a sample proportion.
Step 3, Do: show the calculation
Write the formula, then the formula with your numbers, then the result. If you use a calculator function such as 1-PropZTest, still write the test statistic and p-value.
The first is the one-proportion test statistic; its standard error uses the null value p0. The second is the two-sample t-interval. Its degrees of freedom come from technology, or you can use the smaller of n1 − 1 and n2 − 1 as a conservative choice[3][4].
Step 4, Conclude: in context, linked to the evidence
A conclusion has three parts: the link (“Because the p-value of 0.012 is less than α = 0.05…”), the decision (“…we reject H0“) and the meaning in context (“There is convincing evidence that the true proportion of all students at this school who support a later start time is greater than 0.50”).
With a large p-value you “fail to reject H0” and say there is “not convincing evidence” for Ha. Never say the null is proved or accepted[1]. Figure 1 shows the p-value for the first model answer: the area beyond the observed z, assuming H0 is true.
Model answer 1: a one-proportion z-test
A high school has 2,400 students. The student council takes a simple random sample of 200 students; 116 support a later start time. Is there convincing evidence that more than half of all students at the school support it? (The figures are illustrative.)
Note the details: the parameter names the population, the 10% check uses the population size, the large-counts check uses p0 (a test assumes H0), and the conclusion reuses the context words.
Model answer 2: a two-sample t-interval
Independent random samples of students from two schools record hours of sleep on the last school night. School A (900 students): n1 = 35, x̄1 = 7.4, s1 = 1.6. School B (1,100 students): n2 = 40, x̄2 = 6.6, s2 = 1.9. Construct and interpret a 95% confidence interval for the difference in means. (The figures are illustrative.)
0 is plausible, and saying so is part of full credit. Always state the order of subtraction.
Worked example 3: repairing a weak answer
Student answer to Model answer 1: “H0: p̂ = 0.5, Ha: p̂ > 0.5. SRS, normal. z = 2.26, p = 0.012. Reject H0, so we have proved that most students want a later start.” Fix it.
The repair adds about 40 words. Those words carry the credit.
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A template you can memorise
Fill the brackets with words from the question; the course description recommends using the question’s own language[1].
| Part | Sentence starter |
|---|---|
| Parameter | “Let p (or μ) = the true [proportion / mean] of [variable] for all [population].” |
| Hypotheses | “H0: p = [value]. Ha: p [<, > or ≠] [value]. α = 0.05.” |
| Procedure | “[One / two]-sample [z / t] [test / interval] for [a proportion / a mean / a difference in …].” |
| Random | “The data come from [a random sample of … / random assignment of … to treatments].” |
| 10% | “[n] ≤ 0.10 × [N], so [n] is less than 10% of all [population].” |
| Normality | “[np0] and [n(1 − p0)] are both at least 10″ or “n = [ ] ≥ 30”. |
| Test conclusion | “Because the p-value of [ ] is [less / greater] than α = [ ], we [reject / fail to reject] H0. There [is / is not] convincing evidence that [Ha in context].” |
| Interval conclusion | “We are [C]% confident that the interval from [a] to [b] captures the true [parameter in context].” |
Six reasons answers lose credit
- Vague parameter: “p = proportion” does not say of whom.
- Statistics in hypotheses: p̂ or x̄ in H0.
- Unnamed procedure: see t-test vs z-test for choosing.
- Label-only conditions: each needs a number or reason.
- No link: compare the p-value with α, or check whether the interval contains the null value.
- Context-free or overstated conclusions: say “convincing evidence”, never “proves” or “accept H0“.
Free help, and what the books add
You can go a long way for free. This post gives you the template and model answers. The free statistics guide covers confidence intervals and hypothesis testing step by step. Before you buy anything, flip through the free sample on the product page.
What the books add is volume and checking. The Statistics Made Simple textbook has 16 chapters, every calculation worked by hand, and built-in z, t, chi-square and F tables. The Statistics Made Simple Practice Questions book has 1,569 questions with complete worked solutions, exam-style questions with mark schemes in AP, Cambridge and IB format, and a full 80-mark mock paper. Writing twenty inference answers and checking each one is how the template becomes automatic.
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Practice questions
EasyQ1. A student writes “H0: x̄ = 50, Ha: x̄ ≠ 50″. What is wrong? Show worked answer
Hypotheses must use the parameter μ: “Let μ = the true mean [variable] for all [population]. H0: μ = 50. Ha: μ ≠ 50.”
EasyQ2. In a random sample of 400 adults in a city, 88 cycle to work at least once a week. Construct a 95% confidence interval for the true proportion. Show worked answer
SE = √(0.22 × 0.78 / 400) = 0.0207. Margin = 1.96 × 0.0207 = 0.0406.
Interval: (0.179, 0.261).
We are 95% confident that the interval from 0.179 to 0.261 captures the true proportion of all adults in the city who cycle to work at least weekly.
MediumQ3. A random sample of 25 batteries has mean life 52.3 hours, s = 6.0 hours; a dotplot shows no strong skew or outliers. Construct a 95% confidence interval for the true mean life. Show worked answer
df = 24, t* = 2.064. SE = 6.0/√25 = 1.2. Margin = 2.064 × 1.2 = 2.477.
Interval: (49.82, 54.78) hours, the plausible values for the true mean life of all batteries of this type.
MediumQ4. A test gives p-value = 0.21 with α = 0.05. A student writes: “We accept H0. The new method has no effect on scores.” Rewrite it. Show worked answer
“Because the p-value of 0.21 is greater than α = 0.05, we fail to reject H0. There is not convincing evidence that the new method changes the true mean score of [students in context].”
ExamQ5. A delivery company claims 30% of its parcels arrive a day early. A random sample of 150 of the thousands of parcels sent last month finds 33 arrived a day early. Is there convincing evidence, at α = 0.05, that the true proportion is less than 30%? [4 marks] Show worked answer
Plan: one-sample z-test for a proportion. Random sample; 150 is under 10% of thousands; np0 = 45 ≥ 10, n(1 − p0) = 105 ≥ 10.
Do: p̂ = 33/150 = 0.22. SE = √(0.30 × 0.70 / 150) = 0.0374. z = (0.22 − 0.30)/0.0374 = −2.14. p-value = 0.016.
Conclude: Because 0.016 < 0.05, we reject H0. There is convincing evidence that the true proportion arriving a day early is less than 30%.
Suggested mark scheme (EconTutorials): 1 mark parameter and hypotheses; 1 mark named procedure with three conditions checked numerically; 1 mark z = −2.14 and p ≈ 0.016; 1 mark decision linked to α with conclusion in context.
Frequently asked questions
Do I have to write the hypotheses in words as well as symbols?
You must define the symbol in words, in context. Once p is defined as “the true proportion of all students at this school who…”, H0: p = 0.50 is clear.
Can I use calculator output instead of the formula?
Yes, but name the procedure and write the test statistic, df and p-value or interval.
Why does large counts use p0 for a test but p̂ for an interval?
A test asks what would happen if H0 were true, so it uses p0. An interval has no null value, so it uses the observed successes and failures.
What changes for Cambridge 9709 or IB answers?
The logic is the same: hypotheses about a parameter, the right distribution, a comparison with the significance level and a conclusion in context. The wording and the distributions used can differ, but the four-part structure still keeps your answer complete.
References
- College Board (2026). AP Statistics Course and Exam Description, Effective Fall 2026. New York: College Board. apcentral.collegeboard.org
- Starnes, D. S. and Tabor, J. (2018). The Practice of Statistics, 6th edition. New York: Bedford, Freeman & Worth (Macmillan Learning). ISBN 978-1-319-11333-9.
- Illowsky, B. and Dean, S. (2023). Introductory Statistics 2e, Section 10.1: Two Population Means with Unknown Standard Deviations. Houston, TX: OpenStax, Rice University. openstax.org
- Moore, D. S., McCabe, G. P. and Craig, B. A. (2021). Introduction to the Practice of Statistics, 10th edition. New York: W. H. Freeman / Macmillan Learning. ISBN 978-1-319-24444-6.
