Choose and compute correct t-test

An ads experiment yields two independent samples of watch time per impression, measured in seconds:

• Group A: $n_A = 1200$, $\bar{x}_A = 62$, $s_A = 30$
• Group B: $n_B = 1180$, $\bar{x}_B = 65$, $s_B = 31$ Treat the departure from normality as mild and rely on the large sample sizes. Tasks:

1. Write the null and alternative hypotheses formally, and say whether a one-tailed or two-tailed test is appropriate when the product objective is to detect an increase; explain your reasoning.
2. Calculate the Welch t-statistic, the approximate degrees of freedom, the p-value, and the 95% confidence interval for $\mu_B - \mu_A$. Present the formulas and numeric results rounded to three decimals.
3. Compute Cohen's d using the unbiased pooled standard deviation and describe what the effect size means.
4. Suppose you had pre-registered a one-tailed test for an increase. Recompute the p-value for that specification and discuss the risks of getting the one-tailed direction wrong.
5. Power/MDE: with $\alpha = 0.05$ (two-sided) and power $= 0.8$, what per-group sample size is required to detect a 2-second difference if the common SD is about 30.5? Show the formula and the resulting number. Overview: This question tests a candidate's ability to perform statistical inference for experiments: stating hypotheses and choosing directionality, computing Welch's t-test for independent samples, constructing confidence intervals, calculating Cohen's d, and carrying out power/sample-size calculations.