Compute and interpret 95% confidence intervals

Work through every part. Show the formulas and intermediate calculations. A) An A/B test tracks conversion: Variant A records 410 conversions among 5000 visitors, while Variant B records 470 conversions among 5100 visitors. Calculate a two-sided 95% confidence interval for $p_B - p_A$ using (i) the normal approximation with an unpooled standard error and (ii) the Wilson score (Newcombe) method. For each method, state whether B differs statistically from A at $\alpha=0.05$. Then, if the real setup involved testing three variants (B, C, D) against A with a family-wise $\alpha=0.05$ controlled by Bonferroni, what per-comparison confidence level would you apply, and would the normal-approximation conclusion for B change? B) For average session time in minutes, a sample of $n=40$ gives a mean of 8.2 and a sample standard deviation of 2.4. Compute the 95% confidence interval for the population mean with the t-distribution, and report the margin of error using $t_{0.975,39}$. C) What minimum per-group sample size is needed to estimate a single proportion at 95% confidence so that the interval half-width is no more than 0.01, assuming the conservative $p=0.5$? Give the formula and the result. Overview: This question tests skill with confidence intervals, hypothesis tests for proportions and means, Bonferroni multiple-comparison adjustment, and sample size calculations, including judging the normal approximation against the Wilson score method.