Sample Size for Comparing Two Means

To detect a difference $\delta$ between two means with common standard deviation $\sigma$ , the per-group sample size for a two-sided test is

n = \frac{2\,(z_{1-\alpha/2} + z_{1-\beta})^2\,\sigma^2}{\delta^2}

rounded up. Implement n_per_group(delta, sigma, alpha=0.05, power=0.8), computing normal quantiles yourself by bisection on $\Phi$ (via math.erf).

Also implement mde(n, sigma, alpha=0.05, power=0.8): the minimum detectable effect for a given n per group (the same formula solved for $\delta$ ).