Statistical investigations, simulation and probability decisions

Lesson sequence

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1. Construct three views of one dataset

   Lesson details and objectives

   Build a dotplot, histogram and boxplot and document the choices that make their meanings reproducible.

   • Construct a dotplot and a fully labeled histogram from raw values.
   • Build a boxplot using a declared quartile and whisker convention.
   • Diagnose bin-boundary, density and outlier-display errors.

   6 practice tasks.

2. Compare shapes before choosing summaries

   Lesson details and objectives

   Use supplied observations and constructed displays to compare skew, center, spread and unusual values in context.

   • Compare distributions with common-scale displays.
   • Choose and justify robust or nonrobust summaries for contrasting shapes.
   • Explain how unusual values and sampling scope affect a contextual conclusion.

   6 practice tasks.

3. Check a normal model against raw observations

   Lesson details and objectives

   Estimate center/spread with technology, inspect fit and qualify a normal-area calculation.

   • Calculate a sample mean and sample standard deviation from raw data.
   • Assess a normal approximation using shape, unusual values and sampling context.
   • Use a supplied normal table or technology to estimate a population percentage with limitations.

   6 practice tasks.

4. Calculate and interpret Pearson correlation

   Lesson details and objectives

   Use a reproducible technology workflow for nontrivial Pearson correlation and explain its limits.

   • Calculate Pearson correlation from an entered paired dataset and verify intermediate sums.
   • Interpret correlation with a scatterplot, units and contextual limits.
   • Diagnose constant-variable, unusual-point and nonlinear cases.

   6 practice tasks.

5. Compare fitted models and plotted residuals

   Lesson details and objectives

   Fit three candidate families, compare errors on the same response scale and inspect residual structure.

   • Compare linear, quadratic and exponential fits using the same observed responses.
   • Construct residual plots and diagnose systematic patterns.
   • Interpret contextual coefficients and restrict predictions to justified domains.

   6 practice tasks.

6. Develop a margin from repeated samples

   Lesson details and objectives

   Calibrate an empirical error margin with repeated random samples and test it on separate trials.

   • Develop a finite-simulation margin from absolute sampling errors.
   • Evaluate a fixed calibrated margin using independent simulation draws.
   • Connect a survey interval to its sampling assumptions and distinguish Monte Carlo variability from survey uncertainty.

   6 practice tasks.

7. Simulate treatment reassignment under a null

   Lesson details and objectives

   Build a sharp-null assignment distribution and compare a seeded approximation with exact enumeration.

   • Simulate the original random assignment mechanism under a sharp null.
   • Record assignments, test statistics and a prespecified tail rule.
   • Compare Monte Carlo and exact reference results while limiting causal and population claims.

   6 practice tasks.

8. Count ordered and unordered selections

   Lesson details and objectives

   Define equally likely outcomes before choosing permutations or combinations.

   • Use permutations for distinct roles and combinations for unordered groups.
   • Calculate compound-event probabilities with an explicit denominator.
   • Diagnose ordering, replacement and unequal-selection errors.

   6 practice tasks.

9. Map outcomes to a probability distribution

   Lesson details and objectives

   Map elementary outcomes to a numerical variable and graph its probability law.

   • List elementary outcomes and map each to a value of a random variable.
   • Aggregate repeated values into a normalized mass distribution and graph.
   • Compute and interpret an expectation including a net-payoff transformation.

   6 practice tasks.

10. Build a forecast from observed frequencies

    Lesson details and objectives

    Turn observed frequencies into an inspectable forecasting model and identify its limits.

    • Construct an empirical mass table and graph from observed counts.
    • Calculate an empirical expectation and distinguish it from a known generating law.
    • Evaluate sample scope, missing outcomes and changing conditions before forecasting.

    6 practice tasks.

11. Compare uncertain strategies and consequences

    Lesson details and objectives

    Compare expected consequences while preserving uncertainty, thresholds and risk preferences.

    • Calculate and compare several strategy expectations from an explicit payoff table.
    • Find probability or cost thresholds where the preferred strategy changes.
    • Use conditional risk and consequence data without reducing every decision to expectation alone.

    6 practice tasks.

12. Design and audit a fair random allocation

    Lesson details and objectives

    Design equal-chance procedures and account for rejection, finite limits and implementation bias.

    • Construct an equal-probability allocation from a specified random source.
    • Prove fairness including rejected draws, ties and bounded failures.
    • Audit the difference between equal chances, equal realized counts and other fairness goals.

    6 practice tasks.