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Mathematics · unit version 1

Inference workflows: calculate, check and conclude

Complete proportion, mean and categorical analyses using local numerical reference distributions. Choose methods from design, inspect assumptions and interpret uncertainty; tool output does not certify the study or award mastery.

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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.

Why must the alternative be chosen before seeing the estimated effect?

Choosing a favorable tail after inspecting data changes the claimed error control.

Compare reasoning and find a refresher

A test evaluates a prespecified claim under its null model. Choose less, greater or different from the question and study plan, not from the observed sign.

Review hypotheses and tails

Are two measurements on each participant two independent samples?

Dependence determines the observational unit and standard error.

Compare reasoning and find a refresher

No. For a paired design, form each complete pair’s difference in a declared order, then analyze the mean and sample SD of those differences. Independent-group Welch inference answers a different design question.

Review method and design selection

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Choose and defend an unfamiliar inference workflow

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Lesson sequence

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  1. Complete one- and two-proportion analyses

    Lesson details and objectives

    Keep interval and test standard errors separate while calculating reference probabilities and contextual conclusions.

    • Select one- or independent two-proportion inference from the sampling unit and target parameter.
    • Calculate normal intervals and z tests with estimated versus null-based standard errors.
    • Check counts and design before interpreting intervals, p-values and invalid analyses.

    6 practice tasks.

  2. Complete one-mean, paired and Welch analyses

    Lesson details and objectives

    Choose the independent unit before calculating t probabilities and confidence intervals from summary statistics.

    • Select one-sample, paired-difference or independent-group Welch inference from the study design.
    • Compute t statistics, degrees of freedom, critical values and intervals from appropriate summaries.
    • Interpret evidence while diagnosing missing pair information, zero spread, outliers and unsupported causal claims.

    6 practice tasks.

  3. Calculate categorical evidence and identify its study design

    Lesson details and objectives

    Build expected counts and interpret chi-square tests as association or distribution comparisons according to sampling.

    • Distinguish independence from homogeneity by how units and groups were sampled.
    • Calculate expected cells, chi-square, degrees of freedom and upper-tail probability from a count table.
    • Audit expected counts, unique-unit contributions and the limits of an omnibus conclusion.

    6 practice tasks.