Introduction: Why the t-Test Matters in AP Statistics
In AP Statistics, you’ll often be asked to test hypotheses about means. When the population standard deviation (σ) is unknown — which is almost always in real-world data — you use a t-test instead of a z-test.
The t-test appears in both Multiple Choice (MCQs) and Free Response Questions (FRQs).
It’s essential for:
- One-sample mean inference.
- Two-sample mean comparisons.
- Paired data (before-and-after studies).
This guide — plus RevisionDojo’s t-test worksheets, calculator guides, and FRQ practice banks — will show you how to confidently apply the t-test on the AP exam.
Step 1: What is the t-Distribution?
The t-distribution is similar to the normal distribution, but with:
- Wider tails (accounts for extra uncertainty).
- Degrees of freedom (df): based on sample size (n – 1 for one-sample t-test).
👉 As sample size increases, the t-distribution approaches the normal curve.
Step 2: When to Use a t-Test
You use a t-test when:
- Population mean is unknown.
- Population standard deviation (σ) is unknown.
- You have sample data with mean (x̄), sample standard deviation (s), and sample size (n).
👉 If σ is known (rare), you’d use a z-test.
Step 3: Steps for a Hypothesis Test (t-Test Framework)
Every t-test follows the 4-step process (always write this on FRQs):
- State
- Identify parameter (µ).
- Write null (H₀) and alternative (Hₐ) hypotheses.
- Plan
- Choose correct test (one-sample, two-sample, or paired t-test).
- Check conditions:
- Random sample.
- Normal population OR large n (CLT).
- Independence (10% condition if sampling without replacement).
- Do
- Calculate test statistic:
t=xˉ−μ0s/nt = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
- Find p-value using calculator or t-table.
- Conclude
- Compare p-value to significance level (α = 0.05).
- State conclusion in context.
Step 4: One-Sample t-Test Example
A company claims their light bulbs last 1000 hours. A random sample of 25 bulbs has x̄ = 980, s = 50. Does this suggest the mean life is less than 1000?
- State
- H₀: µ = 1000
- Hₐ: µ < 1000
- Plan
- One-sample t-test. Conditions: random, n = 25 (large enough).
- Do
t=980−100050/25=−2010=−2.0t = \frac{980 - 1000}{50 / \sqrt{25}} = \frac{-20}{10} = -2.0
df = 24 → p ≈ 0.028
- Conclude
Since p < 0.05, reject H₀. Evidence suggests mean life < 1000 hours.
👉 RevisionDojo has worked examples like this with step-by-step solutions.
Step 5: Two-Sample t-Test
Used to compare means from two independent groups.
Example: Compare average test scores between two schools.
Formula:
t=xˉ1−xˉ2s12n1+s22n2t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
👉 On AP exam, use calculator function: 2-SampTTest.
Step 6: Paired t-Test
Used when data comes from the same group measured twice (before-and-after studies).
Example: Same students tested before and after tutoring.
Steps:
- Compute differences (after – before).
- Perform a one-sample t-test on differences.
👉 Paired t-tests are common FRQs since they require careful setup.
Step 7: Using Calculators for t-Tests
TI-84 Commands:
T-Test(one-sample).2-SampTTest(independent samples).TInterval(confidence intervals).
Desmos / StatCrunch:
- Online versions of t-tests.
- Easy to visualize distributions.
👉 RevisionDojo provides calculator walkthroughs with screenshots for t-tests.
Step 8: Interpreting Results
Always interpret results in context of the problem.
Bad: “Reject H₀ because p < 0.05.”
Good: “There is sufficient evidence at the 5% level to conclude that the tutoring program increases average scores.”
👉 Context = key to earning FRQ points.
Step 9: Common Mistakes Students Make
- ❌ Forgetting to check conditions.
- ❌ Using σ instead of s in formula.
- ❌ Mixing up two-sample and paired t-tests.
- ❌ Forgetting conclusion in context.
- ❌ Using calculator output without explanation.
Step 10: t-Test Questions on the AP Stats Exam
MCQ Example
Sample mean = 50, n = 16, s = 4. H₀: µ = 52. What’s the test statistic?
t=50−524/16=−21=−2t = \frac{50 - 52}{4/\sqrt{16}} = \frac{-2}{1} = -2
Answer: –2.
FRQ Example
“Students tested effect of fertilizer on plant growth. 10 plants before treatment, 10 after. Conduct a paired t-test at α = 0.05.”
- Must show:
- Hypotheses.
- Conditions.
- Test statistic + p-value.
- Conclusion in context.
👉 RevisionDojo’s FRQ banks provide worked solutions.
Step 11: Study Hacks for Mastering t-Tests
- Memorize 4-step inference process.
- Always write conditions.
- Practice with calculator functions.
- Redo past FRQs until comfortable.
- Use RevisionDojo’s inference flowcharts to pick the right test quickly.
Step 12: The RevisionDojo Advantage
RevisionDojo helps students master t-tests with:
- Step-by-step inference guides.
- Calculator tutorials (TI-84 + Desmos).
- FRQ practice banks with rubrics.
- Flowcharts for choosing the right test.
👉 Check out RevisionDojo’s AP Stats Inference Resources here.
Frequently Asked Questions (FAQs)
Q: How do I know whether to use a t-test or z-test?
A: Use z-test only if σ is known. Otherwise, use t-test.
Q: What’s the difference between two-sample and paired t-tests?
A: Two-sample = independent groups. Paired = matched or repeated measurements.
Q: Do I need to memorize t-tables?
A: No — you only need to understand interpretation. Calculators handle p-values.
Q: How many points is a t-test FRQ worth?
A: Usually 4–6 points depending on completeness (hypotheses, conditions, calculations, conclusion).
Q: How does RevisionDojo help with t-test mastery?
A: With worksheets, FRQ practice, and calculator guides.
Final Thoughts
The t-test is one of the most practical tools in AP Statistics. You’ll use it in real-world problems, and it’s tested heavily on the AP exam.
To succeed:
- Memorize the 4-step process.
- Know when to use one-sample, two-sample, and paired t-tests.
- Interpret results in context.
- Use RevisionDojo’s resources for guided practice.
With enough practice, you’ll handle any t-test problem with confidence — and score higher on the AP exam.
