Courses / Mathematics / Statistical inference: estimates, tests and defensible conclusions Mathematics · unit version 1 Statistical inference: estimates, tests and defensible conclusions Eighteen lessons develop proportion, mean and categorical inference with design checks, supplied reference values and written interpretation. Examples are original synthetic data. Public practice and report completion are not independently graded mastery or complete AP Statistics certification; complex surveys, regression inference and advanced resampling need further study.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
How do random selection and random assignment support different conclusions? An inference procedure cannot repair a sampling or treatment design that does not support its conclusion.
Compare reasoning and find a refresher Random selection supports population generalization under the sampling design. Random assignment supports causal treatment comparisons under assumptions about implementation and interference. A design can have one without the other.
Review design and confounding
For independent observations with SD 10, what is the SD of their mean when n=25? What n would halve it? Inference separates the variability of observations from the uncertainty of an estimated mean.
Compare reasoning and find a refresher The standard error is 10/√25=2. Quadrupling the sample size to 100 halves it to 1, under the same independent finite-variance model.
Review sampling distributions
A sample of 80 contains 12 successes. What are its observed proportion and failure count? Proportion methods need the right denominator and a check of both outcome counts.
Compare reasoning and find a refresher The observed proportion is 12/80=0.15 and there are 68 failures. These counts alone do not establish random sampling or independence.
Review inference readiness
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Inference: choose methods and defend conclusions
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Lesson sequence Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Lesson details and objectives Separate a fixed parameter, a random interval and coverage assumptions.
Interpret nominal confidence through repeated-interval coverage of a fixed parameter. Calculate an interval midpoint, margin and observed simulation coverage. Distinguish uncertainty in a population mean from individual variability and selection bias. 6 practice tasks.
Lesson details and objectives Construct and critique the introductory normal interval.
Construct a normal proportion interval using the estimated standard error. Check observed success/failure counts and sampling assumptions before using the approximation. Explain why sparse counts and clipped endpoints can undermine Wald interval coverage. 6 practice tasks.
Lesson details and objectives Use a conservative planning value and account for rounding and nonresponse.
Derive the normal planning bound for a specified confidence multiplier and margin. Use the conservative variance factor and round the required sample size upward. Distinguish expected recruitment yield from guaranteed responses and bias control. 6 practice tasks.
Lesson details and objectives Define hypotheses, tail direction and a conditional p-value.
State population hypotheses and choose a directional or two-sided alternative before seeing data. Calculate p-values from supplied reference-tail probabilities. Interpret rejection or nonrejection without assigning a probability to the null hypothesis. 6 practice tasks.
Lesson details and objectives Use null-based uncertainty and check its expected counts.
Calculate a one-proportion z statistic using null-based standard error. Check expected null counts and distinguish them from interval count checks. Use a supplied tail probability to make a qualified contextual test conclusion. 6 practice tasks.
Lesson details and objectives Distinguish Type I error, Type II error and power at a specified alternative.
Identify Type I and Type II errors in a specified decision context. Relate alpha, beta and power to repeated-study probabilities at a stated alternative. Explain threshold, sample-size and multiplicity tradeoffs without claiming that both errors disappear. 6 practice tasks.
Lesson details and objectives Interpret effect sizes and intervals against a decision threshold.
Calculate absolute and relative changes with an explicit baseline. Compare an effect interval with zero and a prespecified practical threshold. Distinguish nonsignificance from an equivalence claim requiring its own margin and procedure. 6 practice tasks.
Lesson details and objectives Use separate group variance estimates for an interval.
Construct an unpooled confidence interval for an oriented difference of proportions. Check four observed counts and independence within and between groups. Reverse interval orientation correctly and identify when paired binary data need another method. 6 practice tasks.
Lesson details and objectives Build a common-rate null model and distinguish it from estimation.
Estimate a pooled common probability from success counts under an equality null. Compute the pooled-null z statistic and expected-count checks. Select the appropriate tail and distinguish equality testing from unpooled interval estimation. 6 practice tasks.
Lesson details and objectives Calculate expected counts and nonnegative discrepancy contributions.
Build an independence expected-count table from row and column totals. Calculate cell contributions, the chi-square statistic and table degrees of freedom. Evaluate count/design conditions and interpret an upper-tail association test without claiming causality. 6 practice tasks.
Lesson details and objectives Keep the same table arithmetic while changing the sampling question.
State an equal-distributions hypothesis for separately sampled or assigned groups. Calculate expected counts and degrees of freedom for a homogeneity table. Distinguish an omnibus conclusion from pairwise claims and plan multiplicity-aware follow-up. 6 practice tasks.
Lesson details and objectives Use a t reference and investigate shape, units and independence.
Construct a one-mean t interval using sample spread and n minus one degrees of freedom. Assess independence, population shape and outliers, especially for small samples. Distinguish a mean-confidence interval from an individual prediction interval. 6 practice tasks.
Lesson details and objectives Interpret a signed t statistic and supplied tail probabilities.
Calculate a signed one-sample t statistic and identify its reference degrees of freedom. Use supplied t-tail probabilities with a prespecified alternative. Explain how outliers, zero sample spread and nonrejection limit a mean-test conclusion. 6 practice tasks.
Lesson details and objectives Preserve matching and define the sign of change.
Define an oriented within-pair difference and compute its mean and standard error. Apply one-sample t inference to independent differences with the correct pair count. Evaluate difference shape, incomplete pairs and limits of causal before-and-after claims. 6 practice tasks.
Lesson details and objectives Use separate spread estimates and an appropriate t reference.
Calculate separate variance contributions and Welch–Satterthwaite degrees of freedom. Construct a two-mean interval and test a zero difference using a supplied critical value. Distinguish independent-group Welch inference from equal-variance pooling and paired analysis. 6 practice tasks.
Lesson details and objectives Match outcome, design and target to an inferential procedure.
Match response type, target parameter and dependence structure to an inference method. Identify the analysis unit and verify design and reference-distribution conditions. Explain when missingness or an unsuitable design prevents a justified calculation. 6 practice tasks.
Lesson details and objectives Enumerate or simulate reassignments that the design actually allowed.
Construct the allowed assignment mechanism under a stated sharp null. Enumerate randomization statistics and calculate an exact tail probability including ties. Explain why blocks, observational confounding and Monte Carlo error constrain a randomization analysis. 6 practice tasks.
Lesson details and objectives Resample units correctly and separate bootstrap uncertainty from bias repair.
Construct same-size resamples with replacement while preserving the correct sampling unit. Interpret a supplied percentile interval and distinguish simulation error from data limitations. Explain why ordinary bootstrap estimation is not automatically a null test or a repair for selection bias. 6 practice tasks.
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