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Inference for categorical data: proportions practice problems AP Statistics, Unit 3

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 free-response question

Confidence intervals and significance tests for one proportion and for the difference of two proportions, with their conditions and interpretation. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Free-response practice
  5. Common mistakes
  6. Keep going

Key equations for inference for categorical data: proportions

p̂ = x / nsample proportion
√(p̂(1 − p̂)/n)standard error of p̂
p̂ ± z* × SEconfidence interval
z = (p̂ − p0) / √(p0(1 − p0)/n)one-proportion z test

The formulas and tables are provided in the exam. Answers within 2.5% are marked correct (5% for probabilities), so values from a table or a calculator both count. The full sheet is in our AP Statistics formula sheet.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
In a random sample of 150 students, 45 walk to school. Find the sample proportion.

Hint

p̂ = successes/sample size.

Worked solution
  1. p̂ = 45/150 = 0.3.
Q2Warm-up
For the same sample, find the standard error of p̂.

Hint

SE = √(p̂(1 − p̂)/n).

Worked solution
  1. SE = √(0.3 × 0.7/150) = √0.0014.
  2. SE = 0.0374.
Q3Warm-up
Which condition checks that the sampling distribution of p̂ is approximately normal?

Hint

This is the Large Counts condition.

Worked solution
  1. At least 10 successes and 10 failures make the normal approximation reasonable.
  2. The answer is C. A and B are the Random and 10% conditions.

Exam-style questions

Questions at the level of the AP Statistics exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
Using the sample of 150 students with p̂ = 0.3, find the margin of error for a 95% confidence interval.

Hint

Margin = z* × SE, with z* = 1.96.

Worked solution
  1. SE = 0.0374.
  2. Margin = 1.96 × 0.0374 = 0.0733.
Q5Exam-style
A coin is tossed 200 times and lands heads 112 times. Find the z test statistic for H0: p = 0.5.

Hint

Use p0 = 0.5 in the standard error, not p̂.

Worked solution
  1. p̂ = 0.56. SE = √(0.5 × 0.5/200) = 0.0354.
  2. z = (0.56 − 0.5)/0.0354 = 1.70.
Q6Exam-style
For that test, with Ha: p > 0.5, find the p-value.

Hint

The p-value is the area to the right of the test statistic.

Worked solution
  1. P(Z > 1.70).
  2. p-value = 0.045.
Q7Exam-style
A test gives a p-value of 0.03. Which is the correct interpretation?

Hint

A p-value is calculated assuming H0 is true.

Worked solution
  1. The p-value is the probability, assuming H0, of a statistic at least as extreme as the one observed.
  2. The answer is C.
Q8Challenge
In two random samples of 100, 60 of the first group and 45 of the second support a proposal. Find the z statistic for H0: p1 = p2.

Hint

Use the pooled proportion in the standard error.

Worked solution
  1. Pooled p̂ = (60 + 45)/200 = 0.525.
  2. SE = √(0.525 × 0.475 × (1/100 + 1/100)) = 0.0706.
  3. z = (0.60 − 0.45)/0.0706 = 2.12.

Free-response practice

One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.

Inference for proportions style · 10 points · about 22 minutes

A town council claims that 40% of residents support a new cycle lane. A local newspaper takes a random sample of 250 residents and finds that 118 support it. Does the sample give convincing evidence that more than 40% support the cycle lane?

  1. (a) State the hypotheses, defining the parameter.
  2. (b) Check the conditions for the test.
  3. (c) Calculate the test statistic and p-value.
  4. (d) State a conclusion in context, at the 5% level.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

p = the proportion of all town residents who support the cycle lane. H0: p = 0.40; Ha: p > 0.40.

Part (b)

Random: random sample. 10%: 250 is less than 10% of the town's residents. Large Counts: 250(0.4) = 100 and 250(0.6) = 150, both at least 10.

Part (c)

p̂ = 0.472. z = (0.472 − 0.40)/√(0.4 × 0.6/250) = 0.072/0.0310 = 2.32. p-value = P(Z > 2.32) = 0.010.

Part (d)

Since 0.010 < 0.05, reject H0. There is convincing evidence that more than 40% of residents support the cycle lane.

Your mark: 0 of 10

Common mistakes in inference for categorical data: proportions

Keep going

Revise the ideas behind these questions:

Then try real exam questions: AP Statistics past papers, with official scoring guidelines. · All AP Statistics units