AP Statistics Unit 5: Statistical Inference
Study confidence intervals, hypothesis tests, p-values, t-tests, chi-square, inference for regression with exam-format practice and rubric-based scoring.
Start AP Stats Practice · Full Study Guide
AP and Advanced Placement are trademarks of College Board. AimFive is not affiliated with or endorsed by College Board.
Inside This Unit: The Full Breakdown
A sampling distribution describes how a statistic (like a sample mean or proportion) varies from sample to sample. The Central Limit Theorem explains why many sampling distributions are approximately normal.
Why it matters
Sampling distributions are the conceptual heart of statistical inference. Every confidence interval and hypothesis test in Units 6-9 relies on understanding how statistics behave across repeated random samples.
Key concepts
- A sampling distribution shows all possible values of a statistic and their probabilities across repeated samples.
- The mean of the sampling distribution of x-bar equals the population mean mu (unbiased).
- The Central Limit Theorem: for large n, the sampling distribution of x-bar is approximately normal regardless of population shape.
- Increasing sample size decreases variability: SD of x-bar = sigma / sqrt(n).
What Is a Sampling Distribution?
Imagine taking every possible sample of size n from a population and computing a statistic (like the mean) for each. The distribution of all those statistics is the sampling distribution. It tells us what values the statistic typically takes, how much it varies, and what shape it follows. The key insight is that statistics are random — different samples produce different values — and the sampling distribution captures that randomness. This concept underlies all inference.
Sampling Distribution of the Sample Mean
If the population has mean mu and standard deviation sigma, then the sampling distribution of x-bar has mean mu and standard deviation sigma/sqrt(n). If the population itself is normal, x-bar is exactly normal for any sample size. If the population is not normal, the Central Limit Theorem says x-bar is approximately normal for sufficiently large n (typically n >= 30). This result is remarkable because it applies regardless of the population shape, enabling inference even when the population distribution is unknown.
Sampling Distribution of the Sample Proportion
For a sample proportion p-hat from a population with proportion p, the sampling distribution has mean p and standard deviation sqrt(p(1-p)/n). When np >= 10 and n(1-p) >= 10, the distribution is approximately normal. This is the foundation for inference about proportions in Units 6 and 8. Larger samples produce smaller standard deviations, making p-hat a more precise estimate of p. The AP exam tests whether students can verify the conditions for normality and correctly compute the standard deviation.
AP exam tip
Always check and state the conditions for using a normal approximation: for means, either the population is normal or n is large (CLT); for proportions, np >= 10 and n(1-p) >= 10.
Connections to other units
- Unit 3: Sampling distributions build on probability concepts and the idea of expected value.
- Unit 6: Confidence intervals for proportions use the normal sampling distribution of p-hat.
- Unit 7: Confidence intervals and tests for means use the t-distribution, which adjusts for estimating sigma.
