Every unit below starts with dead-simple numbers, walks through the formula by hand, then does the exact same calculation on a TI-84. At the end, the easily-confused terms get sorted into one table, and a 10-question quiz checks whether it stuck. Goal: a 5 on the exam.
A z-score tells you how many standard deviations a value is from the mean: z = (x − mean) / standard deviation. Once you have z, you can get a probability instantly from the standard normal table or the calculator's normalcdf.
Use the binomial model when trials are independent, each outcome is success/failure, and the success probability p is the same every trial. The probability of exactly x successes in n trials is P(X=x) = C(n,x)·pˣ·(1−p)ⁿ⁻ˣ.
The distribution of all possible sample means x̄ is called the sampling distribution. The Central Limit Theorem says that as sample size n grows, this distribution gets closer to normal, and its standard deviation (the standard error) shrinks toward σ/√n — bigger samples give more stable sample means.
A confidence interval estimates a plausible range for a parameter; a hypothesis test turns "could this just be chance?" into a probability, the p-value. Significance level α is the threshold you set before the test (usually 0.05); if p-value < α, you reject the null.
When you don't know the population SD σ and estimate it with the sample SD s instead, use the t-distribution (df = n−1) in place of z. The t-distribution has heavier tails than z, so a given statistic produces a slightly larger p-value under t than it would under z.
Measures how far observed counts of a categorical variable are from expected counts. χ² = Σ (observed−expected)² / expected. Bigger differences push χ² up and the p-value down.
Tests whether there's a linear relationship (slope ≠ 0) between two quantitative variables. Divide the slope by its standard error to get a t-statistic, with df = n − 2 (data points minus the 2 estimated parameters: slope and intercept).
These are the concepts most students mix up on the AP exam. Nail them down here, once.
| Term | Definition | Who sets it | How to remember it |
|---|---|---|---|
| α (significance level) | The threshold you're willing to accept for a Type I error. Usually 0.05. | Chosen by the researcher before the test. | "The risk line I'm willing to accept" |
| p-value | Assuming H₀ is true, the probability of getting a result as extreme or more extreme than what was observed. | Calculated from the test statistic. | "How surprising is my data?" |
| Type I Error | Rejecting H₀ when it's actually true. Probability = α. | — | "Convicting an innocent person" |
| Type II Error | Failing to reject H₀ when it's actually false. Probability = β. | — | "Letting the real culprit go free" |
| β | Probability of a Type II error. Larger sample size → smaller β. | Determined by sample size and effect size (not set directly). | "The probability of missing it" |
| Power | Probability of correctly rejecting H₀ when Hₐ is true. Power = 1 − β. | — | "The strength to catch a real effect" |
| z-test | Used when the population SD σ is known. Proportion tests are always z. | — | "Know σ → use z" |
| t-test | Used when σ is unknown and estimated with sample s. Mean tests are almost always t. | — | "Don't know σ → use t (df=n−1)" |
| Binomial | n independent trials, constant success probability p, success/failure outcomes. | — | μ=np, σ=√(np(1−p)) |
| Normal Distribution | Continuous data forming a symmetric bell shape around the mean. Also the limiting shape from the CLT. | — | z = (x−μ)/σ |
Pick an answer and you'll immediately see whether it's correct, plus an explanation. When you're done, hit [Grade Quiz] to see your score.