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Credit Sprint's Guide to AP® Statistics, Unit 4: Inference for Quantitative Data: Means

AP® weight 10–20%

Unit 4 repeats the logic of inference for a quantitative variable. You use the sampling distribution of a sample mean, the central limit theorem and the t distribution to build intervals and run tests for one mean, a mean difference from paired data and the difference between two independent means.

Bigger samples, narrower sampling distributions small n large n center = p or μ (unbiased) SD of p̂ = √[p(1−p)/n] · SD of x̄ = σ/√n

What you need to know

  • The sampling distribution of x̄ has mean μ and standard deviation σ ÷ √n. If the population is normal, the sampling distribution is normal. If not, the central limit theorem says it is approximately normal when n is large (a common guideline is n ≥ 30).
  • When σ is unknown, use the sample standard deviation s and the t distribution with n − 1 degrees of freedom. The t distribution is symmetric and bell-shaped with heavier tails than the normal; it approaches the normal as df grows.
  • A one-sample t interval is x̄ ± t* · s ÷ √n. Conditions: random sample, the 10% condition, and a nearly normal population, which you can justify with n ≥ 30 or a graph with no strong skew or outliers for small samples.
  • A one-sample t test uses t = (x̄ − μ₀) ÷ (s ÷ √n) with df = n − 1. The p-value comes from the t distribution. Conclude in context, linking the p-value to α and to the alternative hypothesis.
  • Matched pairs data (the same subject measured twice, or two similar subjects matched) are analyzed by taking the differences and doing a one-sample t procedure on those differences. State μ_d as the true mean difference.
  • For two independent means, the standard error is √[s₁²/n₁ + s₂²/n₂]. The two-sample t interval is (x̄₁ − x̄₂) ± t* · SE, and the two-sample t statistic is t = [(x̄₁ − x̄₂) − 0] ÷ SE. Degrees of freedom come from technology.
  • Conditions for two-sample means: two independent random samples or two randomly assigned treatment groups, each group less than 10% of its population, and each group approximately normal (or large). Do not use two-sample t on paired data.
  • Interpret the results in context. A confidence interval for a difference that contains 0 does not provide convincing evidence of a difference. An interval for a mean difference that is entirely above 0 supports the claim that the first mean is larger.
  • The width of a t interval depends on the confidence level, the sample standard deviation and the sample size. Larger samples and lower confidence levels give narrower intervals.

Exam traps

Watch forMatched pairs data are not two independent samples. If the same students take a before and after test, use a paired t procedure.
Watch forThe central limit theorem is about the sampling distribution of x̄, not about the data. A skewed sample stays skewed no matter how large n is.
Watch forUse t, not z, when you estimate σ with s.
Watch forA confidence interval is about a mean, not about individual values. Most individual values lie outside a 95% interval for the mean.
Watch forSay 'mean' in the conclusion: 'convincing evidence that the true mean ... is ...', not 'the average person'.

Key terms

  • Sampling distribution of x̄: The distribution of sample means from all possible samples of size n; mean μ, SD σ ÷ √n.
  • Central limit theorem: For a large sample size, the sampling distribution of x̄ is approximately normal, regardless of the population shape.
  • t distribution: A symmetric, bell-shaped distribution with heavier tails than the normal, used when σ is estimated by s.
  • Degrees of freedom (t): For a one-sample procedure, n − 1; for two samples, found with technology.
  • One-sample t interval: x̄ ± t* · s ÷ √n, an interval for a population mean.
  • One-sample t test: Tests a claim about a mean using t = (x̄ − μ₀) ÷ (s ÷ √n).
  • Normality condition (means): The population is approximately normal, or n is large (about 30), or the data show no strong skew or outliers.
  • Matched pairs t procedure: A one-sample t procedure on the differences between paired measurements.
  • Mean difference μ_d: The true mean of the paired differences.
  • Two-sample t procedure: A procedure that compares the means of two independent groups.
  • Standard error of x̄₁ − x̄₂: √(s₁²/n₁ + s₂²/n₂).
  • Independent samples: Samples where the individuals in one group have no connection to the individuals in the other.
  • Paired versus independent: If the same individuals (or matched pairs) are measured twice, use a paired procedure instead of two-sample t.
  • Effect of sample size on width: Larger n gives a smaller standard error and a narrower interval.
  • Effect of confidence level on width: A higher confidence level uses a larger critical value and produces a wider interval.

Practice questions

Five of the 83 Unit 4 questions in Credit Sprint. Every choice is explained in the app.

1. The sampling distribution of x̄ for samples of size n from a population with mean μ and standard deviation σ has mean
  1. σ/√n
  2. σ
  3. x̄
  4. μ
Show answer

μ. The sample mean is an unbiased estimator, so the mean of its sampling distribution equals the population mean.

2. As the degrees of freedom increase, the t distribution
  1. becomes more skewed
  2. approaches the standard normal distribution
  3. becomes wider
  4. shifts to the right
Show answer

approaches the standard normal distribution. With more data, s estimates σ more accurately, so the t curve approaches the normal curve.

3. A 95% confidence interval for μ is (8.2, 9.8). What is the sample mean?
  1. 8.2
  2. 9.0
  3. 0.8
  4. 1.6
Show answer

9.0. The sample mean is the midpoint of the interval: (8.2 + 9.8) ÷ 2 = 9.0.

4. In a matched pairs t-test, what does μ_d represent?
  1. The difference between two sample means
  2. The standard deviation of the differences
  3. The true mean of the differences between paired measurements
  4. The mean of the first group
Show answer

The true mean of the differences between paired measurements. The paired procedure treats each difference as a single value and makes inferences about the mean difference.

5. Which situation requires checking that both samples are less than 10% of their populations in a two-sample t procedure?
  1. Comparing before and after scores
  2. Randomly assigning subjects to treatments
  3. Sampling without replacement from finite populations
  4. Using a very large sample
Show answer

Sampling without replacement from finite populations. The 10% condition helps ensure independence of observations within each sample.