Last Minute AP Statistics Cheat Sheet (WITH FORMULAS)

What You Need to Know

This is the high-yield formula + procedure sheet you use when you’re trying to (1) pick the right method fast, (2) check conditions correctly, and (3) write the minimum-necessary but full-credit inference “story” (parameter → conditions → compute → conclude in context).

Big AP Stats idea: almost every FRQ is either describing data, probability/random variables, or inference (confidence interval or significance test). The fastest way to lose points is skipping conditions or failing to define the parameter.

Golden rule: Your hypotheses and conclusion must be about a population parameter (like $p$, $\mu$, $\mu_1-\mu_2$, $\beta$), not about sample statistics (like $\hat p$, $\bar x$).

Step-by-Step Breakdown

A. Picking the right inference procedure (fast decision tree)

1. Is your response variable categorical (yes/no) or quantitative (number)?
   • Categorical → proportions, chi-square.
   • Quantitative → means, t-procedures, regression.
2. How many groups/samples?
   • 1 sample → one-proportion $z$ or one-mean $t$.
   • 2 independent samples → two-proportion $z$ or two-sample $t$.
   • Matched pairs → one-sample $t$ on differences.
3. Are you comparing distributions of categories across groups?
   • One categorical variable vs a claimed model → chi-square GOF.
   • Two categorical variables (relationship) → chi-square independence.
   • Several populations/treatments and one categorical response → chi-square homogeneity.
4. Is it a relationship between two quantitative variables?
   • Use linear regression; inference about slope $\beta$ uses a $t$ test/interval with $df=n-2$.

B. Writing any inference solution (full-credit skeleton)

1. Define the parameter (in context).
   • Example: $p$ = true proportion of all students at your school who…
2. State hypotheses (test only).
   • $H_0: p=p_0$, $H_a: p\ne p_0$ (or $<$, $>$)
3. Check conditions (name + verify with given info).
   • Random, 10% condition, Normal/Large Counts, etc.
4. Compute the test statistic or interval (show formula + plug values).
5. P-value OR critical value method (usually P-value on AP).
6. Conclude in context at level $\alpha$.
   • “Because $p\text{-value} < \alpha$, reject $H_0$. There is convincing evidence that …”

C. Mini worked walkthrough (one-proportion $z$ test)

Prompt style: “Is there evidence the true proportion differs from $0.40$?”

1. Parameter: $p$ = true proportion of (population) who …
2. Hypotheses: $H_0: p=0.40$, $H_a: p\ne 0.40$
3. Conditions:
   • Random: stated random sample/assignment.
   • 10%: $n \le 0.1N$ (if sampling without replacement).
   • Large counts: $np_0\ge 10$ and $n(1-p_0)\ge 10$.
4. Compute:
   • $\hat p = \dfrac{x}{n}$
   • $z=\dfrac{\hat p - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$
5. Get P-value from Normal.
6. Conclude in context.

Key Formulas, Rules & Facts

A. Describing data (quick hits)

B. Linear transformations & combining variables

C. Probability essentials

D. Discrete random variables (AP favorites)

E. Normal + sampling distributions

F. Confidence intervals (CI) and test statistics (most-used)

One proportion

Two proportions (independent)

One mean (quantitative)

Two means (independent samples)

Matched pairs (paired data)

• Compute differences $d_i = x_{1i}-x_{2i}$.
• Then do one-sample $t$ on differences:
  • $\bar d \pm t^*\dfrac{s_d}{\sqrt{n}}$ and $t=\dfrac{\bar d-\mu_{d,0}}{s_d/\sqrt{n}}$ with $df=n-1$.

G. Chi-square procedures

H. Regression (least squares + inference)

Regression conditions (LINER): Linear pattern, Independent, Normal residuals, Equal variance, Random.

I. Inference vocabulary (quick definitions)

• P-value: probability (assuming $H_0$ true) of getting a statistic as extreme or more extreme than observed.
• Type I error: reject true $H_0$ (false positive). Probability $=\alpha$.
• Type II error: fail to reject false $H_0$ (false negative). Probability $=\beta$.
• Power: $1-\beta$.

Examples & Applications

Example 1: Two-proportion $z$ interval (wording trap)

Situation: Compare vaccination rates in School A vs School B.

• Parameter: $p_A-p_B$ = true difference in vaccination proportions.
• Use CI:
  • $\left(\hat p_A-\hat p_B\right) \pm z^*\sqrt{\dfrac{\hat p_A(1-\hat p_A)}{n_A}+\dfrac{\hat p_B(1-\hat p_B)}{n_B}}$
    Key insight: If CI contains $0$, a “difference” claim isn’t supported.

Example 2: Matched pairs vs two-sample (super common)

Situation: Same students take a pretest and posttest.

• Don’t do two-sample $t$.
• Compute $d_i=post-pre$, then one-sample $t$ on $\mu_d$.
• Test statistic: $t=\dfrac{\bar d-0}{s_d/\sqrt{n}}$.
  Key insight: Pairing reduces variability; ignoring pairing can hide effects.

Example 3: Chi-square independence (interpretation)

Situation: Is seat location (front/middle/back) related to passing (yes/no)?

• Parameter: whether the two categorical variables are independent in the population.
• Expected count: $E=\dfrac{(row\ total)(col\ total)}{n}$.
• Statistic: $\chi^2=\sum \dfrac{(O-E)^2}{E}$, $df=(r-1)(c-1)$.
  Key insight: A significant result says “associated,” not “causes.”

Example 4: Regression slope inference (what you conclude)

Situation: Predict exam score from hours studied.

• Test $H_0: \beta=0$ vs $H_a: \beta>0$.
• Compute $t=\dfrac{b}{SE_b}$ with $df=n-2$.
• Conclusion in context: “There is convincing evidence of a positive linear relationship between hours studied and mean exam score.”
  Key insight: You’re making a claim about mean response changing with $x$, not about individual predictions being perfect.

Common Mistakes & Traps

1. Mistake: Hypotheses about $\hat p$ or $\bar x$ instead of $p$ or $\mu$.

   • Why wrong: sample stats are random; parameters are fixed truths.
   • Fix: define parameter first, then write $H_0$ and $H_a$ about it.

2. Mistake: Using $t$ vs $z$ incorrectly.

   • Why wrong: means with unknown $\sigma$ require $t$; proportions use $z$.
   • Fix: quantitative → $t$, categorical → $z$.

3. Mistake: Pooling in a two-proportion CI.

   • Why wrong: pooling assumes $p_1=p_2$, which is exactly what you’re estimating in a CI.
   • Fix: Pool only for the hypothesis test of $p_1-p_2=0$.

4. Mistake: Skipping or mis-checking large counts.

   • Why wrong: Normal approximation can fail badly with small expected successes/failures.
   • Fix: For one-prop tests use $p_0$; for intervals use $\hat p$.

5. Mistake: Treating matched pairs as independent samples.

   • Why wrong: within-person pairing creates dependence; you must analyze differences.
   • Fix: If the same subject is measured twice (or paired units), do one-sample $t$ on $d$.

6. Mistake: Wrong chi-square df / expected counts.

   • Why wrong: df controls the reference distribution; wrong df → wrong P-value.
   • Fix: GOF $df=k-1$; two-way tables $df=(r-1)(c-1)$; compute $E$ using row/col totals.

7. Mistake: Regression conclusion implies causation.

   • Why wrong: observational studies can have confounding.
   • Fix: Only randomized experiments justify cause-and-effect.

8. Mistake: “No significance” = “proved equal.”

   • Why wrong: failing to reject $H_0$ means insufficient evidence, not proof.
   • Fix: say “not enough evidence to conclude…”

Memory Aids & Quick Tricks

Quick Review Checklist

• [ ] You defined the parameter (with population + context).
• [ ] Your $H_0$ and $H_a$ are about the parameter, and direction matches the prompt.
• [ ] You checked Random, 10%, and the correct Normal/Large Counts condition.
• [ ] Proportions: $z$ procedures; Means: $t$ procedures; Paired: analyze differences.
• [ ] Two-prop test uses pooled $\hat p$; two-prop CI does not.
• [ ] You used the correct df: $n-1$ (one-sample/paired $t$), $n-2$ (regression slope), $(r-1)(c-1)$ (chi-square table).
• [ ] Your conclusion is in context and matches the decision: reject vs fail to reject.
• [ ] You didn’t claim causation unless it was a randomized experiment.

You’ve got this—run the checklist on every inference question and you’ll avoid the biggest point leaks.