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

• Traffic counts
  • Weekday vehicles/day: 80,000 (Mon–Fri)
  • Weekend vehicles/day: 50,000 (Sat–Sun)
  • Average occupants per vehicle: 1.4
• 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)
• Audience segments (share of traffic by passes)
  • Commuters (≥4 weekdays): 50%
  • Locals (within 3 km): 30%
  • Occasional passers: 20%
• Pass frequency per person by segment (weekly)
  • Commuters: Poisson(λ=5)
  • Locals: Poisson(λ=2)
  • Occasional: Poisson(λ=1)
• 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

1. 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).
2. 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.
3. Identify at least three major sources of bias (e.g., panel selection, deduplication error, dwell-time bias) and propose corrections/validations.