Estimate total attendance from size-biased reservation sample

You operate a restaurant that handles N = 10,000 reservations over the course of a day. For each reservation j, you have:

• Reserved party size: $R_j$ (a positive integer)
• Actual number of people who arrive: $S_j$ (which may be less than, equal to, or greater than $R_j$) To cut down on measurement work, you only record $(R,S)$ for a sample of n = 1,000 reservations. The sampling is size-biased:
• A reservation with reserved size 10 is twice as likely to enter the sample as a reservation with reserved size 5.
• More generally, suppose 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 you need, and explain how you would quantify uncertainty (for example, a confidence interval). Overview: This question tests understanding of size-biased sampling, selection bias correction, finite-population total estimation, and uncertainty quantification in survey sampling.