Hypothesis testing: a two-sample t-test from scratch

The problem, from first principles

Group A (a new website layout) averages 52 seconds on-page, group B (the old layout) averages 48 seconds. Is the new layout genuinely better, or would two random samples from the SAME underlying population, with no real difference at all, plausibly show a 4-second gap just from ordinary sampling noise? 01-confidence-interval quantified uncertainty around ONE group's mean; a hypothesis test directly answers the "is this difference real" question for TWO groups, by asking exactly how surprising the observed gap would be if there were actually no true difference at all.

This is the formal machinery behind essentially every "does this change actually help" decision made with data: an A/B test comparing two website variants, a clinical trial comparing a drug to a placebo, an ML experiment comparing a new model architecture against a baseline, they're all, underneath, this exact same two-sample comparison.

From theory to code

Theory builds Welch's t-test (the version that doesn't assume equal variance between groups, the safer, more commonly recommended default) from two pieces: a t-statistic measuring how many standard errors apart the two means are, and the Welch-Satterthwaite equation for the (non-integer) effective degrees of freedom needed to convert that statistic into a p-value.

Implement welch_t_statistic(a, b) and welch_degrees_of_freedom(a, b) first, then two_sample_t_test(a, b), which combines them into a (t_statistic, p_value) pair.

Constraints

• Do not assume equal variance between a and b (this is specifically Welch's t-test, not the classic Student's equal-variance version).
• two_sample_t_test computes a two-sided p-value.
• Use scipy.stats.t.cdf for the p-value lookup, not a hand-rolled approximation.

Hints

Open one at a time. Each gives away a little more than the last.