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).