Explain BLS vs CLS; compute t-stats

Part A — Concepts: Explain the difference between a Brand Lift Study (BLS) and a Conversion Lift Study (CLS) in advertising measurement. For each method, identify the main sources of bias and variance (for example, survey non-response bias, contamination, auction interference, and selection). Discuss situations where BLS and CLS could produce conflicting results and how you would bring them into agreement. Give a short comparison of difference-in-differences (DiD) and propensity score matching (PSM) for observational advertising data; state the identifying assumptions for each approach and one robustness check for each.

Part B — Hypothesis test for proportions (CLS): In a randomized CLS holdout, the treatment arm has $n_1=1200$ users and $x_1=42$ conversions; the control arm has $n_0=1180$ users and $x_0=33$ conversions. Calculate: (1) the point lift ($p_1 - p_0$), (2) the pooled standard error under $H_0: p_1 = p_0$, and (3) the two-sided t/z-statistic and p-value. Indicate whether the effect is significant at $\alpha=0.05$.

Part C — Manual DiD t-statistic (BLS survey): You measure Purchase Intent (binary) before and after for exposed and control cohorts: Group sizes: $n_{T,pre}=300$, $n_{T,post}=320$, $n_{C,pre}=290$, $n_{C,post}=310$. Sample means: $m_{T,pre}=0.18$, $m_{T,post}=0.26$, $m_{C,pre}=0.17$, $m_{C,post}=0.20$. Sample variances (Bernoulli sample variances): $s^2_{T,pre}=0.1476$, $s^2_{T,post}=0.1924$, $s^2_{C,pre}=0.1411$, $s^2_{C,post}=0.1600$. Tasks: (1) Compute the DiD estimator: $(m_{T,post}-m_{T,pre})-(m_{C,post}-m_{C,pre})$. (2) Assuming independent samples and unequal group sizes, derive and compute an approximate standard error for the DiD by summing the four independent sample variances scaled by their respective sample sizes, and then (3) compute the t-statistic and state your conclusion at $\alpha=0.05$. Show the formulas you use explicitly.

Overview: This question tests causal inference and experimental measurement abilities in digital advertising, covering the differences between Brand Lift Studies and Conversion Lift Studies, bias and variance sources, observational identification approaches (difference-in-differences and propensity score matching), and applied hypothesis testing for proportions and DiD variance estimation. It is often used to assess whether a candidate can reason through the trade-offs between survey-based and experiment-based metrics, state identifying assumptions and robustness checks, and carry out practical statistical inference; the domain is Statistics & Math for a data scientist role, and the level covers both conceptual understanding and hands-on application.