AP Statistics

306 core topics · approximately 98 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph - topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. Mixed checks follow every 6 lessons; each unit ends with a 12-item quiz, and course-wide assessments appear at 25%, 50%, 75%, and 100% mastery. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

AP Statistics: Exploring One- & Two-Variable Data · 12 topics

Mean, Median & Mode [M]	Three measures of center - and when each is the right one.
Range, IQR & Standard Deviation [M]	Spread: from the crude range to the resistant IQR to standard deviation.
Five-Number Summary & Boxplots [M]	Min, Q1, median, Q3, max - the skeleton of a boxplot.
z-Scores & Percentiles [M]	Standardize a value into standard-deviation units.
Describing Distributions [M]	Shape, center, spread - and always in context.
Outliers & Linear Transformations [M]	Flag outliers with 1.5·IQR; predict how rescaling shifts summaries.
The Empirical Rule (68-95-99.7) [M]	Bell curves keep 68%, 95%, 99.7% within 1, 2, 3 standard deviations.
Two-Way Tables: Marginal & Conditional [M]	Read joint, marginal, and conditional proportions from a table.
Correlation: Interpreting r [M]	What the correlation coefficient does - and does not - tell you.
Least-Squares Regression Line [M]	Build and use ŷ = a + bx to predict.
Residuals [M]	Residual = observed − predicted; the leftover a line misses.
r² - the Coefficient of Determination [M]	r² is the fraction of variation the line explains.

AP Statistics: Collecting Data · 11 topics

Population vs Sample; Parameter vs Statistic [M]	Who we study versus who we measure; the truth versus our estimate.
Sampling Methods [M]	SRS, stratified, cluster, and systematic sampling.
Sources of Bias [M]	How samples systematically miss the truth.
Observational Study vs Experiment [M]	Watching versus doing - and what each can conclude.
Confounding Variables [M]	A lurking cause that mimics or masks the effect you care about.
Random Assignment vs Random Selection [M]	One earns causation; the other earns generalization.
Experimental Design: Control & Treatment [M]	Control groups, treatments, placebos, and comparison.
Blocking & Matched Pairs [M]	Group similar units first, then randomize within - like stratifying.
Generalizability & Scope of Inference [M]	How far your conclusions legitimately reach.
Explanatory & Response Variables [M]	Which variable predicts, which one you measure.
Principles of Experimental Design [M]	Control, randomization, replication - the three pillars.

AP Statistics: Probability & Random Variables · 13 topics

Basic Probability Rules [M]	Favorable over total; every probability lives in [0, 1].
The Complement Rule [M]	P(not A) = 1 − P(A); the fast route to 'at least one'.
The Addition Rule [M]	Add the pieces, then subtract the double-counted overlap.
Conditional Probability [M]	Probability once you restrict to a sub-population.
Independence & the Multiplication Rule [M]	Independent events multiply.
Expected Value of a Discrete RV [M]	The probability-weighted average outcome.
Variance & SD of a Discrete RV [H]	Average squared distance from the mean of a random variable.
Combining Independent Random Variables [H]	Means always add; variances add for independent variables.
Binomial Probability [H]	Count the arrangements, multiply the per-path probability.
Binomial Mean & Standard Deviation [M]	μ = np and σ = √(np(1 − p)).
Geometric Probability [M]	Waiting for the first success.
Normal Probabilities & Percentiles [M]	Turn Normal questions into z-scores and back.
Normal Probabilities with a Table [H]	Standardize to a z-score, then read the area from the Normal table.

AP Statistics: Statistical Inference · 11 topics

Sampling Distribution of x̄ [M]	Same center as the population, but less spread.
Sampling Distribution of p̂ [M]	Centered at p, spread √(p(1 − p)/n).
The Central Limit Theorem [M]	Big samples make x̄ Normal, whatever the population's shape.
Standard Error Formulas [M]	Estimate how much a statistic bounces from sample to sample.
Margin of Error [M]	How far the estimate can plausibly be from the parameter.
Confidence Intervals [M]	estimate ± margin of error.
Interpreting Confidence Intervals [M]	Confidence is about the method, not one computed interval.
Null & Alternative Hypotheses [M]	State the 'no effect' claim and the effect you're testing for.
Hypothesis Test Logic & Decisions [M]	Compare the p-value to α; reject or fail to reject.
Interpreting a p-value [M]	A p-value assumes the null - it is not P(null is true).
Type I & II Errors and Power [M]	Two ways to be wrong, and the chance of being right.

AP Statistics: t-Procedures, Two-Sample & χ² Inference · 12 topics

The t-Distribution & Degrees of Freedom [M]	When σ is unknown you use t, not z - and df drives its shape.
One-Sample t Interval for a Mean [H]	Estimate μ with x̄ ± t*·(s/√n) when σ is unknown.
One-Sample t Test for a Mean [H]	Test H₀: μ = μ₀ with t = (x̄ − μ₀)/(s/√n).
One-Proportion z Test [H]	Test H₀: p = p₀ with z = (p̂ − p₀)/√(p₀(1 − p₀)/n).
Two-Proportion z Interval & Test [H]	Compare two proportions: interval unpooled, test pooled.
Two-Sample t Interval for a Difference of Means [H]	Estimate μ₁ − μ₂ with (x̄₁ − x̄₂) ± t*·√(s₁²/n₁ + s₂²/n₂).
Two-Sample t Test for a Difference of Means [H]	Test H₀: μ₁ = μ₂ with t = (x̄₁ − x̄₂)/√(s₁²/n₁ + s₂²/n₂).
Matched-Pairs t Procedures [H]	Paired data → one-sample t on the differences.
Chi-Square Goodness-of-Fit Test [H]	Compare observed counts to expected with χ² = Σ(O − E)²/E, df = k − 1.
Chi-Square Test for Independence / Homogeneity [H]	Two-way table: expected = row×col/total, df = (r − 1)(c − 1).
Inference for a Regression Slope [H]	Test and estimate β with t = b/SE_b on df = n − 2.
Checking Conditions for Inference [M]	Random, 10%, and Normal/Large-Counts before every procedure.

AP Statistics: Data, Design & Probability - Depth · 45 topics

The Mean [H]	Add the values, divide by how many there are.
The Median [H]	The middle value once the data are sorted.
Mode & Range [H]	The most frequent value; the full span of the data.
Quartiles & the IQR [H]	Split the sorted data into quarters; the middle half is the IQR.
Sample Standard Deviation [H]	Typical distance from the mean, dividing squared deviations by n − 1.
Population Standard Deviation [H]	Same idea as s, but dividing squared deviations by n.
Variance [H]	The square of the standard deviation.
Five-Number Summary [H]	Min, Q1, median, Q3, max - the skeleton of a boxplot.
Outliers: the 1.5·IQR Rule [H]	Flag values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.
z-Scores (Standardizing) [H]	How many standard deviations a value sits from the mean.
The Empirical Rule (68-95-99.7) [H]	Normal curves keep 68%, 95%, 99.7% within 1, 2, 3 SDs.
Linear Transformations of Data [H]	Adding shifts center only; multiplying scales center and spread.
Weighted Mean [H]	Average where some values count more than others.
Mean from a Frequency Table [H]	Weight each value by its frequency, then divide by the count.
Shape & Skewness [H]	Skew drags the mean toward the tail; report resistant summaries.
Resistant vs Non-Resistant Measures [H]	Median and IQR shrug off outliers; mean and SD do not.
Percentiles [H]	The percent of values at or below a given value.
Interpreting Correlation r [H]	r measures the strength and direction of a linear trend.
Properties of r [H]	Unitless, bounded by ±1, unchanged by shifting or rescaling.
LSRL Slope from r, sₓ, s_y [H]	Slope b = r·(s_y/sₓ).
LSRL Intercept [H]	The line passes through (x̄, ȳ), so a = ȳ − b·x̄.
Prediction with the LSRL [H]	Plug an x into ŷ = a + bx to get the predicted y.
Residuals [H]	Residual = observed − predicted.
r² - Coefficient of Determination [H]	Square the correlation to get the fraction of variation explained.
Interpreting r² as a Percent [H]	r² is the percent of variation in y the model accounts for.
Extrapolation [H]	Predicting outside the data range is risky.
Reading Residual Plots [H]	No leftover pattern means a line is a good fit.
Population, Sample, Parameter, Statistic [H]	Who we study vs who we measure; the truth vs our estimate.
Sampling Methods [H]	SRS, stratified, cluster, and systematic sampling.
Sources of Bias [H]	How samples systematically miss the truth - and a bigger n won't fix it.
Observational Study vs Experiment [H]	Watching vs doing - and what each can conclude.
Confounding Variables [H]	A lurking cause that mimics or masks the effect you care about.
Random Assignment vs Random Selection [H]	One earns causation; the other earns generalization.
Experimental Design: Control & Blinding [H]	Control groups, placebos, and double-blind comparison.
Blocking & Matched Pairs [H]	Group similar units first, then randomize within - like stratifying.
Principles of Experimental Design [H]	Control, randomization, replication - the three pillars.
Explanatory & Response Variables [H]	Which variable predicts, which one you measure.
Basic Probability Rules [H]	Favorable over total; every probability lives in [0, 1].
The Complement Rule [H]	P(not A) = 1 − P(A).
The Addition Rule [H]	Add the pieces, then subtract the double-counted overlap.
Conditional Probability [H]	Probability once you restrict to a sub-population.
Independence & the Multiplication Rule [H]	Independent events multiply.
Mutually Exclusive vs Independent [H]	Disjoint events can't overlap; independent events don't influence.
Expected Value [H]	The probability-weighted average outcome.
'At Least One' Probabilities [H]	P(at least one) = 1 − P(none).

AP Statistics: Distributions & Inference - Depth · 45 topics

Mean of a Discrete Random Variable [H]	μ_X = Σ x·P(x), the probability-weighted average.
Variance of a Discrete Random Variable [H]	σ²_X = Σ (x − μ)²·P(x).
SD of a Discrete Random Variable [H]	σ_X = √(variance), back in the original units.
Linear Transformation of a Random Variable [H]	Y = a + bX shifts and scales the mean; only |b| scales the SD.
Combining Means of Random Variables [H]	Means always add (or subtract): E(X ± Y) = E(X) ± E(Y).
Combining Variances of Independent RVs [H]	For independent X, Y variances ADD for both sums and differences.
Combining Standard Deviations [H]	SDs do NOT add - add variances first, then square-root.
Binomial Probability P(X = k) [H]	P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ.
Binomial Mean μ = np [H]	The expected number of successes is np.
Binomial Standard Deviation [H]	σ = √(np(1 − p)).
Binomial 'At Least One' [H]	P(X ≥ 1) = 1 − P(X = 0) = 1 − (1 − p)ⁿ.
Geometric Probability [H]	P(X = k) = (1 − p)ᵏ⁻¹·p - waiting for the first success.
Geometric Mean 1/p [H]	On average it takes 1/p trials to reach the first success.
Normal Percentages (Empirical Rule) [H]	Use 68-95-99.7 and symmetry to get common Normal percents.
Standardizing a Normal Value [H]	z = (x − μ)/σ turns any Normal value into a standard score.
Normal Value from a Percentile [H]	Invert with x = μ + z·σ for the common empirical percentiles.
The Standard Normal Distribution [H]	N(0, 1): the yardstick every Normal is measured against.
Center of the Sampling Distribution of x̄ [H]	x̄ is unbiased: μ_x̄ = μ.
Standard Error of x̄ [H]	σ_x̄ = σ/√n - bigger samples, less variability.
Center of the Sampling Distribution of p̂ [H]	p̂ is unbiased: μ_p̂ = p.
Standard Error of p̂ [H]	σ_p̂ = √(p(1 − p)/n).
The Central Limit Theorem [H]	Large samples make x̄ approximately Normal, whatever the population shape.
The Large Counts Condition [H]	np ≥ 10 and n(1 − p) ≥ 10 make p̂ approximately Normal.
Normal Condition for Means [H]	Normal population or a large sample (CLT) justifies t/z for a mean.
Bias vs Variability [H]	Center handles bias; sample size handles variability.
Effect of Sample Size [H]	SE ∝ 1/√n, so quadrupling n halves the standard error.
z Confidence Interval for a Mean [H]	x̄ ± z*·(σ/√n) when σ is known.
Margin of Error for a Mean [H]	Margin of error = critical value × standard error.
Critical Values z* [H]	1.645 → 90%, 1.96 → 95%, 2.576 → 99%.
Confidence Interval for a Proportion [H]	p̂ ± z*·√(p̂(1 − p̂)/n).
Standard Error for a Proportion Interval [H]	In a CI, estimate the SE with p̂: √(p̂(1 − p̂)/n).
Margin of Error for a Proportion [H]	ME = z*·√(p̂(1 − p̂)/n).
Interpreting a Confidence Interval [H]	Confidence is about the method's long-run capture rate, not one interval.
What Changes Interval Width [H]	Higher confidence widens; larger samples narrow.
Setting Up Hypotheses [H]	H₀ is the status quo; both hypotheses are about parameters.
One-Proportion z Statistic [H]	z = (p̂ − p₀)/√(p₀(1 − p₀)/n).
One-Sample z Statistic for a Mean [H]	z = (x̄ − μ₀)/(σ/√n) when σ is known.
Interpreting a p-Value [H]	P(data this extreme | H₀ true) - small means evidence against H₀.
The Decision Rule [H]	Reject H₀ when p-value ≤ α; otherwise fail to reject.
Type I & Type II Errors [H]	Type I: reject a true H₀ (rate α). Type II: keep a false H₀.
Choosing t vs z [H]	Means with unknown σ → t; proportions → z.
Chi-Square Expected Counts [H]	Expected = (row total × column total) / grand total.
Chi-Square Degrees of Freedom [H]	GOF: df = k − 1. Two-way: df = (r − 1)(c − 1).
Chi-Square Contributions [H]	Each cell contributes (O − E)²/E; sum them for χ².
Computing the Chi-Square Statistic [H]	χ² = Σ (O − E)²/E summed over every category.

AP Statistics - Deep II · 12 topics

Center of the Sampling Distribution of x̄₁ − x̄₂ [H]	The difference of sample means is centered at the difference of the means.
Standard Error of x̄₁ − x̄₂ [H]	Variances add: the SE of a difference is √(σ₁²/n₁ + σ₂²/n₂).
Normal Probabilities for a Difference of Means [H]	Standardize the difference by its standard error, then use the 68-95-99.7 rule.
Center of the Sampling Distribution of p̂₁ − p̂₂ [H]	The difference of sample proportions is centered at p₁ − p₂.
Standard Error of p̂₁ − p̂₂ [H]	Add the two sampling variances p(1−p)/n, then take the square root.
Power as 1 − β [H]	Power is the chance of correctly rejecting a false null: power = 1 − β.
Factors That Affect Power [H]	Bigger samples, larger α, and larger effects all raise power.
Sample Size for a Proportion [H]	Solve MOE = z*√(p(1−p)/n) for n, using p = 0.5 to be safe.
Sample Size for a Mean [H]	Solve MOE = z*σ/√n for n: n = (z*σ / MOE)².
Interpreting a Confidence Level [H]	The confidence level describes the long-run capture rate of the method.
Interpreting the Margin of Error [H]	The margin of error is the interval's reach on each side of the estimate.
Standard Error vs. Standard Deviation [H]	SD describes spread of data; SE describes spread of a statistic.

AP Statistics - Exploring One and Two Variable Data (Deep) · 50 topics

Describing a Distribution Completely [H]	A complete description reports shape, center, spread, and unusual features in context.
Shape from a Dotplot [H]	Read shape from a dotplot by where the long tail points and how many peaks appear.
Gaps, Clusters and Isolated Values [H]	Gaps, clusters, and isolated values are unusual features that must be reported.
Inferring Shape from Summary Statistics [H]	Compare the mean with the median and the two tail lengths to infer skew.
Locating the Median in a Histogram [H]	Accumulate the frequencies until you pass half the observations to find the median's interval.
Reading a Cumulative Relative Frequency Plot [H]	A cumulative relative frequency plot gives the percent at or below each value, so subtract to get an interval.
Comparing Centers of Two Distributions [H]	Compare two distributions by reporting the difference in their centers with units and context.
Comparing Spreads of Two Distributions [H]	Compare variability with the IQR, the resistant measure of spread read from a boxplot.
Writing a Comparison in Context [H]	A comparison must use comparative language, cite both centers and both spreads, and stay within what the data support.
What Boxplots Cannot Show [H]	A boxplot shows the five-number summary only, so it hides sample size, modes, the mean, and the standard deviation.
How an Outlier Moves the Mean [H]	Recompute a mean after a value is added by rebuilding the total, since the mean is not resistant.
How Little an Outlier Moves the Median [H]	The median depends only on position, so adding one extreme value shifts it at most one position.
Recomputing a Mean After Removing Values [H]	Recover the total from n times the mean, subtract the removed values, then divide by the count that remains.
Outliers, the Standard Deviation and the IQR [H]	Squared deviations make the standard deviation sensitive to outliers, while the IQR ignores them.
Comparing Performances with z-Scores [H]	Standardizing removes units and centers, so the larger z-score marks the better relative performance.
Solving for a Missing Mean or Standard Deviation [H]	Rearrange z = (x - mu)/sigma to recover whichever of mu or sigma is unknown.
The Empirical Rule on Measurement Values [H]	Convert each boundary to a whole number of standard deviations, then combine the 68-95-99.7 areas.
Expected Counts from the Empirical Rule [H]	Multiply the empirical-rule percent for a region by the sample size to get an expected count.
Normal Areas from Boundaries [H]	Standardize each boundary, then subtract cumulative Normal areas to get the proportion between them.
Normal Boundaries from Areas [H]	Find the z-score with the required area below it, then unstandardize with x = mu + z sigma.
The Middle c Percent of a Normal Model [H]	Split the leftover area between the two tails, so the middle c percent runs from the (50 - c/2) to the (50 + c/2) percentile.
Solving a Normal Model for sigma [H]	Convert the stated area to a z-score, then divide the distance from the mean by that z-score.
Solving a Normal Model for mu [H]	Convert the stated area to a z-score, then subtract z sigma from the given value to recover the mean.
Counting Outliers from Raw Data [H]	Compute the quartiles, build both fences, then count the values outside them.
Building the Fences from Raw Data [H]	The fences sit one and a half IQRs beyond the quartiles, so the quartiles must be found from the data first.
Recovering a Quartile from a Fence [H]	Reverse the 1.5 IQR rule by treating the fence equation as an equation to solve for the unknown quartile.
Reading Percentages from a Boxplot [H]	The quartiles cut a boxplot into four parts holding about 25 percent of the data each.
Form, Direction and Strength [H]	Describe a scatterplot by its form, its direction, and the strength of the pattern, in context.
Matching a Scatterplot to r [H]	The sign of r follows the direction of the pattern and its magnitude follows how tightly the points hug a line.
Why r Alone Cannot Judge Form [H]	The correlation measures only linear association, so a large r never establishes that a line is the right model.
Finding the Point that Departs from the Pattern [H]	The point that departs most from a linear pattern is the one with the largest residual in absolute value.
The Least-Squares Line from Summary Statistics [H]	Build the line from b = r times s_y over s_x and the point of averages, then predict.
Interpreting a Slope in Context [H]	The slope is the predicted change in y for each one-unit increase in x, with the units of y per unit of x.
Interpreting an Intercept in Context [H]	The intercept is the predicted value of y when x equals zero, and it is meaningful only if x = 0 is within the data's reach.
Recovering r from the Slope [H]	Invert b = r times s_y over s_x to get r = b times s_x over s_y.
Inverse Prediction from a Regression Line [H]	Set the predicted value equal to the target and solve the linear equation for x.
Recovering an Observed Value from a Residual [H]	Since residual equals observed minus predicted, the observed value is the prediction plus the residual.
Residuals from a Least-Squares Line Sum to Zero [H]	Least-squares fitting forces the residuals to sum to zero, so a missing residual is the negative of the others' total.
Reading a Residual Plot [H]	A residual plot shows observed minus predicted, so a point below zero marks an over-prediction.
The Sign of a Residual in Context [H]	A negative residual means the model over-predicted and the point lies below the line.
Recovering the Sign of r from r Squared [H]	r squared hides the direction, so take the square root and attach the sign of the slope.
Stating r Squared Precisely [H]	r squared is the percent of the variation in y explained by the linear model, not the percent of points on the line.
What Changes r and What Does Not [H]	The correlation is unitless and unchanged by linear rescaling or by swapping the variables, while the slope is not.
High Leverage, Outliers and Influence [H]	Leverage comes from an extreme x value, an outlier from a large residual, and influence from removing the point changing the fit.
Diagnosing an Influential Point [H]	A point is influential when deleting it markedly changes the slope, the intercept, or r squared.
Refitting the Slope with a Point Removed [H]	Refit the least-squares slope from the remaining points using the sums of the deviation products.
Choosing a Transformation to Linearize [H]	Constant multiplicative growth calls for log y on x, a power relationship for log y on log x, and logarithmic growth for y on log x.
Back Transforming an Exponential Fit [H]	Predict the transformed response first, then undo the logarithm to return to the original units.
Back Transforming a Power Fit [H]	A line fitted to log y against log x is a power model, so substitute log x and then undo the logarithm.
Reading Growth from a Transformed Slope [H]	In a log y model the slope is a multiplicative rate: each unit of x multiplies the predicted y by the base raised to the slope.

AP Statistics - Sampling, Experiments and Probability (Deep) · 48 topics

A Simple Random Sample from a Line of Random Digits [H]	Read equal-length digit groups in order, keeping labels that are in range and skipping repeats.
Proportional Allocation in a Stratified Sample [H]	Proportional allocation gives every stratum the same sampling fraction, so n_i = n(N_i/N).
Why Stratify: Precision, Not Fairness [H]	Stratifying reduces variability when the strata differ from one another and are similar inside.
Cluster Samples versus Stratified Samples [H]	A cluster sample measures every individual in a few randomly chosen groups; a stratified sample takes some individuals from every group.
The Interval in a Systematic Sample [H]	A systematic sample takes every k-th individual with k = N/n after a random start, so the j-th selection is start + (j - 1)k.
Periodicity Breaks a Systematic Sample [H]	When the list repeats with a period matching the sampling interval, every selection shares the same position in the cycle and the sample is biased.
Sampling Frames and Undercoverage [H]	Undercoverage occurs when the list sampled from omits part of the population, giving those individuals no chance of selection.
Nonresponse versus Voluntary Response [H]	Nonresponse is failure to reply by individuals selected at random; voluntary response is self-selection by individuals never sampled.
Question Wording and Response Bias [H]	A leading or loaded question pushes answers in a predictable direction, biasing the estimate that way.
Predicting the Direction of a Bias [H]	A biased design misses the parameter in a predictable direction, and a larger sample does not remove it.
Retrospective and Prospective Studies [H]	Both look at groups the subjects already belong to: a retrospective study reads past records, a prospective study follows subjects forward in time.
Factors, Levels and the Number of Treatments [H]	A treatment is one combination of levels, so the number of treatments is the product of the numbers of levels.
Group Sizes in a Completely Randomized Design [H]	In a completely randomized design every unit is assigned at random among the groups, whose sizes sum to the number of units.
What Replication Means [H]	Replication means applying each treatment to several experimental units, not repeating a measurement or the whole study.
The Placebo Effect and a Placebo Control [H]	The placebo effect is a response to the act of being treated, so a placebo control group is needed to isolate the treatment effect.
Single-Blind and Double-Blind Experiments [H]	An experiment is single-blind when one of the two parties, subjects or measurers, is unaware of the assignment, and double-blind when both are.
Choosing a Blocking Variable [H]	Block on a variable known to affect the response, then randomize treatments within each block.
Randomizing Within Blocks [H]	In a randomized block design each block is split at random among the treatments, so a block of size s gives s/t units per treatment.
Randomization in a Matched-Pairs Design [H]	A matched-pairs design is blocking with blocks of size two: the assignment is randomized within each pair and the paired differences are analyzed.
Scope of Inference: Which Conclusion Is Earned [H]	Random selection permits generalization to the population; random assignment permits a cause-and-effect conclusion.
Diagnosing the Flaw in an Experiment [H]	A sound experiment needs comparison, random assignment and replication; each missing element permits a specific alternative explanation.
Assigning Digits to Model a Probability [H]	Use the shortest digit group whose count of equally likely outcomes matches the probability, and assign that share of the groups to success.
Estimating a Probability from Simulated Trials [H]	A simulation estimates a probability by the proportion of trials in which the event occurred, and a mean by the average trial result.
Choosing a Correct Simulation Setup [H]	A valid simulation matches the probability of each outcome, the number of repetitions in one trial, and whether selection is with or without replacement.
The Addition Rule for Three Events [H]	Add the single probabilities, subtract every pairwise overlap, then add back the triple overlap.
Mutually Exclusive, Independent, or Neither [H]	Mutually exclusive means P(A and B) = 0; independent means P(A and B) = P(A)P(B); events with positive probabilities cannot be both.
The Overlap That Independence Forces [H]	Independence fixes the intersection at P(A and B) = P(A)P(B), which also determines the union.
Reversing a Conditional in a Two-Way Table [H]	The condition sets the denominator: P(row | column) divides by the column total, P(column | row) by the row total.
The Law of Total Probability [H]	Split the sample space into cases, multiply each case probability by its conditional probability, and add.
Reversing a Conditional in a Screening Test [H]	P(condition | positive) is the probability of the true-positive path divided by the total probability of a positive result.
Testing Independence in a Two-Way Table [H]	Two events in a table are independent exactly when the conditional proportion equals the overall proportion.
Completing a Probability Distribution [H]	The probabilities of a discrete random variable sum to 1, which recovers any single missing value.
Standard Deviation from a Probability Table [H]	Compute the mean, then the probability-weighted average squared deviation, then take the square root.
Interpreting the Mean of a Random Variable [H]	The mean of a random variable is its long-run average value, not its most likely value and not necessarily attainable.
Expected Net Gain [H]	Expected net gain is the expected payout minus the fixed cost, and it scales with the number of repetitions.
Tail and Conditional Probabilities from a Table [H]	Add the probabilities of the values in the event; a condition restricts the table and rescales by the probability of the condition.
The Total of n Copies versus n Times One [H]	Variances add for independent copies, so the total of n copies has SD σ√n, while n times a single observation has SD nσ.
Mean and Standard Deviation of aX + bY [H]	Means combine linearly; for independent variables the variances combine as a²σ²_X + b²σ²_Y, whatever the signs.
Checking the Binomial Conditions [H]	A binomial count needs a fixed number of independent trials, two outcomes per trial, and the same success probability throughout.
Binomial Probability over a Range [H]	Add the binomial probabilities of every count in the range, or subtract the complement when fewer terms are needed.
The Most Likely Binomial Count [H]	The most likely count of a binomial variable is the largest whole number not exceeding (n + 1)p.
Cumulative Geometric Probabilities [H]	For a geometric count P(X greater than k) = (1 - p)^k, and the model is memoryless, so past failures do not change what follows.
Binomial or Geometric? [H]	Count successes in a fixed number of trials with the binomial model; count trials up to the first success with the geometric model.
A Probability for a Sample Mean [H]	Standardize with z = (x̄ - μ)/(σ/√n), then read the Normal table.
When the Central Limit Theorem Is Needed [H]	The Central Limit Theorem makes the sampling distribution of x̄ approximately Normal for a large sample, whatever the population shape.
A Probability for a Sample Proportion [H]	Standardize with z = (p̂ - p)/√(p(1 - p)/n), then read the Normal table.
The 10% Condition [H]	When sampling without replacement, the standard error formulas require the sample to be at most one tenth of the population.
The Shape of the Distribution of p-hat [H]	The distribution of p̂ is approximately Normal only when np and n(1 - p) are both at least 10; otherwise it is skewed away from the nearer boundary.

AP Statistics - Statistical Inference (Deep) · 47 topics

Conditions for a Proportion Interval [H]	A one-sample z interval for p requires Random, 10%, and Large Counts.
Constructing a Proportion Interval [H]	The interval is p-hat plus or minus z* times the square root of p-hat(1 minus p-hat)/n.
Interpreting a Proportion Interval [H]	An interval estimates the population proportion, not the sample proportion or individuals.
Recovering p-hat, the Margin of Error and n [H]	The centre of an interval is the estimate and half its width is the margin of error.
Assessing a Claim with a Proportion Interval [H]	Values inside a confidence interval are plausible; values outside it are not.
Conditions for a t Interval for a Mean [H]	A t interval needs Random, 10%, and a Normal population or a large sample.
Constructing a t Interval for a Mean [H]	The interval is x-bar plus or minus t* times s/root n, with df = n minus 1.
Interpreting a t Interval for a Mean [H]	A confidence interval estimates the population mean, not individual values.
Recovering x-bar, the Margin of Error and s [H]	From a reported interval, x-bar is the midpoint and the margin of error is the half-width.
Changing the Confidence Level [H]	With the sample fixed, the margin of error is proportional to z*.
Changing the Sample Size [H]	The margin of error shrinks in proportion to the square root of the sample size.
Comparing Margins of Error [H]	The margin of error rises with z* and with p-hat near 0.5, and falls as n grows.
Precision Against Confidence [H]	At a fixed sample size, more confidence costs precision and less confidence buys it.
Planning a Sample Size from a Pilot Estimate [H]	Solve the margin of error formula for n, using a pilot value of p-hat, and round up.
The Conservative Sample Size [H]	Using p = 0.5 maximizes p(1 minus p) and guarantees the target margin of error.
Planning a Sample Size for a Mean [H]	Solve m = z* sigma over root n for n, giving n = (z* sigma / m) squared, and round up.
How Precision Scales with Sample Size [H]	Dividing the margin of error by k requires multiplying the sample size by k squared.
Hypotheses Stated in Context [H]	The null states no effect at the claimed value; the alternative states the effect asked about.
Parameters, Not Statistics, in Hypotheses [H]	Hypotheses are claims about population parameters, never about sample statistics.
Writing a Conclusion in Context [H]	Compare the p-value with alpha, then state the decision and the evidence in context.
The p-Value as a Conditional Probability [H]	A p-value is the chance of a result this extreme or more, computed assuming the null is true.
Finding a p-Value from a Normal Table [H]	The p-value is the tail area beyond z, doubled for a two-sided alternative.
Misreadings of a p-Value [H]	A p-value measures evidence against the null, not the probability that a hypothesis holds.
Significance, Effect Size and Design [H]	A small p-value shows an effect is detectable, not that it is large or causal.
The One-Proportion z Statistic from Counts [H]	Standardize p-hat using the null value: z = (p-hat minus p0) over root p0(1 minus p0)/n.
Which Proportion Goes in the Standard Error [H]	A test standardizes with the null value p0; an interval must estimate with p-hat.
The Pooled Two-Proportion z Test [H]	Under H0 the two proportions are equal, so the counts are pooled into one estimate.
Conclusions from a Two-Sample Test [H]	Random assignment permits a causal conclusion; random sampling permits generalization.
A t Statistic from Raw Data [H]	From raw data, compute x-bar and s first, then t = (x-bar minus mu0) over s/root n.
The Two-Sample t Statistic [H]	Standardize the observed difference of means about the hypothesized difference.
Paired Data or Two Independent Samples [H]	Data are paired when each observation in one group is naturally linked to one in the other.
Paired t Procedures from a Data Table [H]	Reduce paired measurements to one column of differences and analyse that column.
A Type I Error in Context [H]	A Type I error rejects a null hypothesis that is actually true.
A Type II Error in Context [H]	A Type II error fails to reject a null hypothesis that is actually false.
Controlling Type I and Type II Errors [H]	Lowering alpha reduces Type I errors; only more data reduces both error rates at once.
Power Described in Context [H]	Power is the chance a test detects a specified true alternative value.
Intervals and Two-Sided Tests Agree [H]	A C% interval rejects exactly the null values it excludes, at alpha = 1 minus C.
Reading an Interval for a Difference [H]	An interval for a difference shows a difference only when it excludes zero.
Matching a Level to a Significance Level [H]	A two-sided test at alpha matches a confidence level of 1 minus alpha.
Expected Counts for a Goodness-of-Fit Test [H]	Each expected count is the sample size times the claimed proportion for that category.
The Chi-Square Goodness-of-Fit Statistic [H]	Add (observed minus expected) squared over expected across every category.
The Largest Chi-Square Contribution [H]	A category's contribution compares its squared deviation with its expected count.
Choosing the Right Chi-Square Procedure [H]	One sample and one variable is goodness of fit; one sample and two variables is independence; several samples is homogeneity.
Chi-Square from a Two-Way Table [H]	Each expected count is the row total times the column total divided by the grand total.
A t Statistic for a Regression Slope [H]	The slope statistic is the estimate minus the hypothesized slope, divided by its standard error.
A Confidence Interval for a Slope [H]	The interval for beta is b plus or minus t* times the standard error of the slope, on n minus 2 degrees of freedom.
Conclusions about a Regression Slope [H]	A significant slope shows a linear relationship in the population, not causation or strength.

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