Estimate billboard reach and impressions
Overview
This question evaluates probabilistic modeling and statistical inference skills applied to audience measurement and attribution, covering Poisson-based frequency modeling, visibility-adjusted impressions, panel-based deduplication and expansion, uncertainty propagation (delta method/bootstrap), and Markov-chain attribution within the Statistics & Math / Data Science domain. It is commonly asked to test the ability to convert traffic and visibility inputs into quantitative reach and impressions with propagated confidence intervals, reason about attribution via a 3-state Markov chain, identify measurement biases, and demonstrates both conceptual understanding and practical application through numerical estimation and sensitivity analysis.
Digital Billboard: Weekly Reach, Impressions, and Store-Visit Attribution
Context
You are estimating the performance of a single digital billboard beside a 6‑lane urban expressway over one week. You must report weekly unique reach (people who saw it at least once) and total impressions, then attribute store visits via a simple 3‑state Markov chain. Assume traffic counts refer to passes that could potentially see the billboard (directionality already accounted for), and average vehicle occupancy applies uniformly.
Given Data and Assumptions
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Traffic counts
- Weekday vehicles/day: 80,000 (Mon–Fri)
- Weekend vehicles/day: 50,000 (Sat–Sun)
- Average occupants per vehicle: 1.4
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Visibility probability per pass
- Daytime p_vis_day = 0.75; Night p_vis_night = 0.55
- Daytime traffic share = 70% (thus weekly average p_vis = 0.7×0.75 + 0.3×0.55)
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Audience segments (share of traffic by passes)
- Commuters (≥4 weekdays): 50%
- Locals (within 3 km): 30%
- Occasional passers: 20%
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Pass frequency per person by segment (weekly)
- Commuters: Poisson(λ=5)
- Locals: Poisson(λ=2)
- Occasional: Poisson(λ=1)
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Panel for deduplication
- Weekly unique devices within 500 m: 120,000
- Device-to-person expansion factor: 2.2
- Panel capture-rate uncertainty: ±10% (1σ), to be propagated (delta method or bootstrap)
Tasks
- Estimate weekly unique reach and total impressions with 95% confidence intervals. Clearly state all formulas and independence assumptions. Show how to combine traffic counts, occupancy, visibility, and pass frequency to compute impressions, and how to deduplicate to people-level using the panel (including the capture-rate uncertainty via delta method or bootstrap).
- Using a 3-state Markov chain (Unaware → Aware → Visit), propose reasonable segment-specific transition probabilities and compute expected visits/week attributable to the billboard. Discuss sensitivity to these probabilities and to p_vis.
- Identify at least three major sources of bias (e.g., panel selection, deduplication error, dwell-time bias) and propose corrections/validations.
Solution
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