Inference for Means review games for AP Statistics.
This unit covers t-intervals, one-sample t-test, two-sample t-test and paired t-test — essential concepts for AP Statistics. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers t-intervals, one-sample t-test, two-sample t-test and paired t-test — essential concepts for AP Statistics. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 T-Intervals
A t-interval estimates a population mean when the population standard deviation is unknown, using the sample standard deviation s as a substitute. The critical value comes from the t-distribution with n−1 degrees of freedom, which has heavier tails than the normal distribution. You must verify conditions (Random, Normal/Large Sample, Independent) before constructing the interval.
Key Points
- Formula: x̄ ± t* · (s/√n), where t* is the critical value from t-distribution with df = n−1
- Conditions: Random sample, population distribution normal OR n ≥ 30 (CLT), and n ≤ 10% of population
- Wider interval than z-interval because t* > z* for the same confidence level, especially at small n
- Increasing confidence level widens the interval; increasing n narrows it
A random sample of 16 students has a mean sleep time of 6.8 hours and s = 1.2 hours. Construct a 95% confidence interval for the mean sleep time.
With df = 15, the critical value t* ≈ 2.131. The margin of error is 2.131 · (1.2/√16) = 2.131 · 0.3 ≈ 0.639. The interval is (6.161, 7.439). We are 95% confident the true mean sleep time is between 6.16 and 7.44 hours.
2 One-Sample T-Test
A one-sample t-test determines whether there is convincing statistical evidence that a population mean differs from a claimed value μ₀. The test statistic measures how many standard errors the sample mean lies from the hypothesized mean. You must state hypotheses, check conditions, compute the test statistic and p-value, and write a conclusion in context.
Key Points
- Test statistic: t = (x̄ − μ₀) / (s/√n), compared to t-distribution with df = n−1
- H₀: μ = μ₀; Hₐ can be <, >, or ≠ depending on the research question
- P-value is the probability of obtaining a result as extreme or more extreme assuming H₀ is true
- If p-value < α (usually 0.05), reject H₀ and conclude there is convincing evidence for Hₐ
A cereal company claims boxes contain 18 oz on average. A consumer group samples 20 boxes and finds x̄ = 17.7 oz and s = 0.6 oz. Test whether the true mean weight is less than 18 oz at α = 0.05.
H₀: μ = 18, Hₐ: μ < 18. The test statistic is t = (17.7 − 18)/(0.6/√20) = −0.3/0.134 ≈ −2.24 with df = 19. The p-value ≈ 0.019. Since 0.019 < 0.05, we reject H₀ and conclude there is convincing evidence that the true mean weight is less than 18 oz.
3 Two-Sample T-Test
A two-sample t-test compares the means of two independent populations to determine if there is evidence of a difference. The two samples must be collected independently of each other — this is what distinguishes it from a paired t-test. Degrees of freedom are calculated using technology (or conservatively as the smaller of n₁−1 and n₂−1 by hand).
Key Points
- Test statistic: t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂), df computed by calculator
- H₀: μ₁ − μ₂ = 0 (or μ₁ = μ₂); Hₐ specifies direction of difference
- Conditions must hold for BOTH groups: each sample is random, each is normal/large, and both are independent of each other
- Do NOT pool variances on the AP exam unless explicitly told populations have equal variance
Researchers compare exam scores for students taught by Method A (n=25, x̄=82, s=8) vs. Method B (n=30, x̄=78, s=10). Is there evidence at α = 0.05 that Method A produces higher mean scores?
H₀: μ_A = μ_B, Hₐ: μ_A > μ_B. The test statistic is t = (82 − 78)/√(64/25 + 100/30) = 4/√(2.56 + 3.33) = 4/2.427 ≈ 1.65. Using a calculator, df ≈ 52 and p-value ≈ 0.052. Since 0.052 > 0.05, we fail to reject H₀ — there is not convincing evidence that Method A produces higher scores.
4 Paired T-Test
A paired t-test is used when two measurements come from the same subject or matched pairs, such as before-and-after designs. The key step is computing the difference d = x₁ − x₂ for each pair and treating those differences as a single sample. This reduces variability and is more powerful than a two-sample test when pairing is appropriate.
Key Points
- Reduce to one-sample problem: find d̄ (mean of differences) and s_d (SD of differences), then apply one-sample t-test
- Test statistic: t = d̄ / (s_d/√n), with df = n−1 where n is the number of pairs, not total observations
- Pairing is justified when measurements are naturally linked (same person, matched subjects, repeated measures)
- Confidence interval for mean difference: d̄ ± t* · (s_d/√n)
Eight athletes have their resting heart rate measured before and after a 6-week training program. The differences (before − after) are: 5, 3, 8, 2, 6, 4, 7, 5. Test whether training reduces heart rate at α = 0.05.
Compute d̄ = (5+3+8+2+6+4+7+5)/8 = 40/8 = 5.0 and s_d ≈ 1.93. The test statistic is t = 5.0/(1.93/√8) = 5.0/0.682 ≈ 7.33 with df = 7. The p-value is essentially 0 (much less than 0.05), so we reject H₀ and conclude there is convincing evidence that the training program reduces mean resting heart rate.
Questions, answered.
What is Inference for Means?
Inference for Means is Unit 7 of AP Statistics, covering t-intervals, one-sample t-test, two-sample t-test and paired t-test.
How to study for AP Statistics Unit 7?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.