Compute and correct correlation significance inflation

You computed correlations for sales outreach analysis. Answer the following with formulas and numerical results where possible. (a) For n = 3200 deals, Pearson r between call_count in the first 14 days post-creation and is_won is 0.23. Using Fisher's z-transform, compute the 95% CI for r and the two-sided p-value. Show intermediate z, SE, and back-transform steps. (b) You tested m = 24 correlations (different channels/time windows). The sorted p-values are: [0.0004, 0.0010, 0.0040, 0.0090, 0.0120, 0.0190, 0.0260, 0.0310, 0.0410, 0.0530, 0.0610, 0.0740, 0.0810, 0.0940, 0.1100, 0.1300, 0.1700, 0.2100, 0.2700, 0.3400, 0.4100, 0.5500, 0.6800, 0.7900]. Apply Benjamini–Hochberg at q = 0.10 and state which hypotheses you reject, showing the threshold comparison i*(q/m). (c) What is the minimal detectable correlation (two-sided, alpha = 0.05, power = 0.80) for n = 500 using Fisher's z power approximation? Provide the formula and numeric answer. (d) You observe overall corr(discount_rate, is_won) = -0.10, but within each region {East, West, Central} the correlations are {+0.05, +0.04, +0.03}. Explain, with equations, how region-mix imbalance can yield this Simpson’s paradox and how to diagnose it numerically (e.g., weighted covariance decomposition and partial correlation controlling for region).