Statistical Significance for Group Growth

Apply the addition rule to calculate the probability that at least one of two possibly overlapping events occurs.

Apply the multiplication rule to independent and conditional sequences of events.

Determine whether a repeated-trials scenario satisfies the assumptions of a binomial model.

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

Calculate exact binomial probabilities with scipy.stats.binom.

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

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

Visualize the sampling distribution of a proportion and explain how its spread changes with group size.

Compute and interpret a confidence interval for a single binary-event rate.

Compute and interpret a confidence interval for the difference between two independent binary-event rates.

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

Interpret a p-value as a tail probability under the null hypothesis and identify common misinterpretations.

Perform and interpret an exact binomial test of a rate against a benchmark in SciPy.

Perform and interpret a two-proportion hypothesis test for independent groups.

Quantify the rise in false-positive risk when many hypotheses are tested by using probability calculations and Python simulation.

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

Choose between family-wise error rate and false discovery rate based on the consequences of false discoveries.

Apply the Benjamini–Hochberg procedure to a collection of group-level p-values.

Explain how group size, baseline rate, and effect size influence standard errors, p-values, and statistical power.

Estimate the power of a rate comparison through repeated Python simulation.

Calculate an approximate required sample size or minimum detectable effect for a two-group rate comparison.

Carry out an end-to-end analysis of multiple group rates in pandas and report statistical uncertainty, multiplicity adjustments, and practical significance.