Statistical Significance for Group Growth

Derive the binomial probability mass function from combinations and the multiplication rule.

Calculate point, cumulative, and tail probabilities with scipy.stats.binom.

Simulate binomial experiments with NumPy and compare simulated frequencies with theoretical probabilities.

Derive the expected value, variance, and standard error of a sample proportion under a binomial model.

Use Python simulation to visualize the sampling distribution of a proportion and explain how its spread changes with group size and baseline rate.

Compute and interpret a confidence interval for a single binary-event rate using its repeated-sampling meaning.

Estimate the difference between two independent binary-event rates, compute a confidence interval for it, and assess its practical magnitude.

Formulate null and alternative hypotheses for a benchmark or independent-rate comparison and choose a one- or two-sided test before examining the result.

Locate an observed statistic in an exact or simulated null distribution, interpret its p-value as a tail probability under the null, and identify common p-value misinterpretations.

Perform an exact binomial test of a rate against a benchmark in SciPy and report the estimate, confidence interval, and p-value.

Perform a two-proportion hypothesis test for independent groups, check its approximation conditions, and report the rate difference, confidence interval, and p-value.

Quantify the rise in the probability of at least one false positive as more hypotheses are tested, using probability calculations and Python simulation.

Define a family of hypotheses and choose between family-wise error rate and false discovery rate based on the consequences of false discoveries.

Apply the Bonferroni correction to control the probability of any false positive across a family of tests.

Apply the Benjamini–Hochberg procedure to group-level p-values and interpret the resulting discoveries under false-discovery-rate control.

Predict how group size, baseline rate, and effect size influence standard errors, p-values, and statistical power when other factors are held fixed.

Estimate the power of an independent two-rate comparison through repeated Python simulation and interpret low power as a risk of missing a real effect.

Calculate the approximate per-group sample size required to detect a specified rate difference at chosen significance and power levels.

Carry out an end-to-end analysis of multiple independent group rates in pandas and report effect sizes, confidence intervals, multiplicity-adjusted results, and practical significance.