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Inference for quantitative data: means practice problems AP Statistics, Unit 4

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

t intervals and t tests for one mean, the difference of two means, and paired data, 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 quantitative data: means

SE = s / √nstandard error of x̄
t = (x̄ − μ0) / (s/√n)one-sample t test, df = n − 1
x̄ ± t* × s/√none-sample t interval
SE = √(s12/n1 + s22/n2)two-sample standard error

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
How many degrees of freedom does a one-sample t procedure have for a sample of 25?

Hint

df = n − 1.

Worked solution
  1. df = 25 − 1 = 24.
Q2Warm-up
A sample of 25 has standard deviation 10. Find the standard error of the sample mean.

Hint

SE = s/√n.

Worked solution
  1. SE = 10/√25 = 10/5 = 2.
Q3Warm-up
Why do we use t instead of z for inference about a mean?

Hint

What extra uncertainty does the t distribution account for?

Worked solution
  1. We almost never know σ, so we use s. That adds variability, which the t distribution allows for with heavier tails.
  2. The answer is B.

Exam-style questions

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

Q4Exam-style
A random sample of 36 bags of rice has mean mass 52 g and standard deviation 6 g. Find the t statistic for H0: μ = 50.

Hint

t = (x̄ − μ0)/(s/√n).

Worked solution
  1. SE = 6/√36 = 1.
  2. t = (52 − 50)/1 = 2.
Q5Exam-style
A random sample of 16 has standard deviation 8. Find the margin of error of a 95% t interval for the mean.

Hint

t* has 15 degrees of freedom.

Worked solution
  1. SE = 8/√16 = 2. With df = 15, t* = 2.131.
  2. Margin = 2.131 × 2 = 4.26.
Q6Exam-style
A two-sided t test has t = 2.0 with 35 degrees of freedom. Find the p-value.

Hint

Two-sided: double the tail area.

Worked solution
  1. P(t > 2.0) with df = 35 is about 0.0267.
  2. p-value = 2 × 0.0267 = 0.053.
Q7Exam-style
Twenty students each take a test before and after a revision course. Which procedure compares the scores?

Hint

Are the two sets of scores independent, or linked student by student?

Worked solution
  1. Each student gives two linked measurements, so analyse the 20 differences.
  2. That is a paired t test: B.
Q8Challenge
Group 1: mean 78, SD 10, n = 25. Group 2: mean 72, SD 12, n = 30. Find the two-sample t statistic for the difference in means.

Hint

SE = √(s12/n1 + s22/n2).

Worked solution
  1. SE = √(100/25 + 144/30) = √(4 + 4.8) = 2.966.
  2. t = (78 − 72)/2.966 = 2.02.

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 means style · 10 points · about 22 minutes

A coach records the 100 m times of 10 randomly selected sprinters before and after a six-week training programme. The mean of the differences (before minus after) is 0.18 seconds, with standard deviation 0.20 seconds.

  1. (a) State the hypotheses for testing whether the programme reduces times, defining the parameter.
  2. (b) Name the test and check its conditions.
  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)

μd = the true mean difference (before minus after) for all such sprinters. H0: μd = 0; Ha: μd > 0.

Part (b)

Paired t test. Random: random selection of sprinters. 10%: 10 is less than 10% of all sprinters. Normal: n is small, so check that a graph of the differences shows no strong skew or outliers.

Part (c)

t = 0.18/(0.20/√10) = 0.18/0.0632 = 2.85, df = 9. p-value = P(t > 2.85) = 0.0096.

Part (d)

Since 0.0096 < 0.05, reject H0. There is convincing evidence that the programme reduces 100 m times on average for sprinters like these.

Your mark: 0 of 10

Common mistakes in inference for quantitative data: means

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