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We are asked to compute the sample size manually given: $$ \delta = 0.04, \quad \pi_1 = 0.72, \quad \pi_2 = 0.84, \quad \alpha = 0.02, \quad \beta = 0.25 $$ This looks like a sample size calculation for testing the difference between two proportions. Let’s solve step by step. --- ### Step 1: Identify the formula The standard formula for the sample size per group when comparing two proportions is: $$ n = \frac{(Z_{\alpha/2} \sqrt{2\pi(1-\pi)} + Z_\beta \sqrt{\pi_1(1-\pi_1) + \pi_2(1-\pi_2)})^2}{(\pi_2 - \pi_1)^2} $$ Where: * $\pi = \frac{\pi_1 + \pi_2}{2}$ (pooled proportion) ________ _____ ________ ___ ______.
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