Set the Group Call participant cap

We need to choose a maximum participant limit $K$ for Group Calls. You have call-level telemetry in calls(call_id, start_ts, participants_count, video_on_ratio, median_mos_score, p95_jitter_ms, packet_loss_pct, bitrate_kbps, cpu_load_pct, network_type, region, device_tier, dropped_call_flag). Propose a statistical approach for selecting $K$ that maximizes total successful participant-minutes while satisfying the guardrails that the P95 of median_mos_score is at least $3.8$ and dropped_call_rate is at most $2\%$. Include: (a) how you would fit and validate a piecewise/segmented regression or change-point model relating QoE to participants_count; (b) how you would quantify uncertainty in $K$ and produce a 95% confidence interval within $\pm 1$ participant; (c) whether to use one global $K$ or dynamic caps by device/network, and the rule for collapsing to a single product limit; (d) the number of calls needed to estimate $K$ at that precision and how you would test robustness to heavy-tail outliers. Overview: This question assesses a candidate's skill in applied statistical modeling, change-point or segmented regression, uncertainty quantification, sample-size estimation, and robustness analysis, using telemetry to set operational participant caps while meeting QoE guardrails such as median MOS and dropped-call rate.