Welch's t-Test for Two Means

Did the new checkout flow change average order value? For continuous metrics, compare means with Welch's t-test, which, unlike Student's t-test, doesn't assume the two groups have equal variance.

With sample means $\bar a, \bar b$ , sample variances $s_a^2, s_b^2$ (with ddof=1) and sizes $n_a, n_b$ :

t = \frac{\bar a - \bar b}{\sqrt{s_a^2/n_a + s_b^2/n_b}}, \qquad \nu = \frac{\left(s_a^2/n_a + s_b^2/n_b\right)^2}{\frac{(s_a^2/n_a)^2}{n_a - 1} + \frac{(s_b^2/n_b)^2}{n_b - 1}}

Implement welch_t(a, b) returning (t, df). Raise ValueError if either group has fewer than 2 values. Then implement mean_diff_ci(a, b, z=1.96), an approximate 95% confidence interval for $\bar a - \bar b$ using the same standard error: (low, high).