Estimate total attendance from size-biased reservation sample

You run a restaurant with **N = 10,000 reservations** in a day. Each reservation *j* has: - Reserved party size: \(R_j\) (positive integer) - Actual number of people who show up: \(S_j\) (can be \(<\), \(=\), or \(>\) \(R_j\)) To reduce measurement effort, you only observe \((R,S)\) for a **sample of n = 1,000 reservations**. The sampling is **size-biased**: - A reservation with reserved size 10 is **twice as likely** to be sampled as a reservation with reserved size 5. - More generally, assume the sampling probability is **proportional to reserved size**: \(\Pr(\text{reservation } j \text{ is sampled}) \propto R_j\). ### Task Using the 1,000 sampled pairs \((R_i,S_i)\), estimate **the total number of people who would show up across all 10,000 reservations**: \[ T_S = \sum_{j=1}^{10000} S_j. \] State any assumptions needed, and explain how you would quantify uncertainty (e.g., a confidence interval).