Compute A/B significance, CI, and power

You conduct an A/B test over seven days. Control A records 520 conversions in 10,000 sessions. Variant B records 630 conversions in 11,500 sessions. A calculator may be used as needed. Treat the observations as independent Bernoulli trials. a) Run a two-proportion z-test of $H_0: p_B - p_A = 0$ against a two-sided alternative. Calculate the pooled-proportion z statistic and the corresponding two-sided p-value. b) Using the unpooled standard error, compute a 95% confidence interval for $p_B - p_A$. c) For sample-size planning with baseline $p_0 = p_A$, what equal per-arm sample size $n$ (in sessions) is required to detect an absolute lift of $0.7$ percentage points (that is, $p_1 = p_0$ and $p_2 = p_0 + 0.007$) at a two-sided $\alpha = 0.05$ and 80% power using the normal approximation? Use $z_{0.975} = 1.96$ and $z_{0.80} = 0.84$; state the formula you apply and the resulting $n$ rounded up. d) If you were instead testing 12 independent metrics, what Bonferroni-adjusted per-metric $\alpha$ would keep the family-wise $\alpha_{FWER} = 0.05$? Overview: This question assesses skill in statistical inference for A/B testing: two-proportion z-tests, confidence intervals built with unpooled standard errors, sample-size and power calculations, and Bonferroni correction for multiple comparisons.