Question
A grant agency publishes a spreadsheet with 18 records, one for every application that received funding during the year. Each record includes the grant amount and the successful applicant's details. Applications that were rejected are not stored in this extract, and the number of applications submitted is not reported.
A director asks, “What fraction of all applications submitted this year were funded?” Can the analyst determine that historical fraction from the spreadsheet, even if every published winner is recorded perfectly? Give two different complete application histories consistent with the same spreadsheet but having different success fractions. State the smallest extra type of count that would make the fraction calculable.
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Hint 1
The spreadsheet reveals the numerator, the number of funded applications, but not the total number considered.
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Solution
The empirical funding rate for the year is \(r=\text{funded applications}/\text{total submitted applications}\). We know the numerator is exactly 18, but the denominator \(N\) is absent. The winner-only spreadsheet identifies the amounts and characteristics conditional on being funded; it does not identify \(\Pr(\text{funded})\) across all submissions.
Consider two possible underlying histories:
- History 1: Twenty applications were submitted, 18 funded and two rejected. The published spreadsheet has the same 18 winner records, and the funding fraction is \(18/20=90\%\).
- History 2: Two hundred applications were submitted, 18 funded and 182 rejected. The identical spreadsheet of 18 funded applications is published, but the funding fraction is \(18/200=9\%\).
Nothing in the retained records distinguishes these histories. Many other denominators \(N\ge18\) are likewise possible, yielding \(18/N\); the fraction can be arbitrarily small as \(N\) increases. It is therefore not identifiable from this outcome-truncated extract, regardless of how precisely the funded amounts are measured.
A single extra count, either total applications \(N\) or rejected applications \(R\), is sufficient to compute the historical fraction as \(18/N\) or \(18/(18+R)\). This would be an observed fraction for this year, not proof that a future applicant's underlying success probability is known without sampling uncertainty.
Assumptions
Each application corresponds to at most one funded record and all 18 funded applications are included exactly once.
Rejected applications do not appear in the spreadsheet, and no additional source reveals the total number submitted.
The requested estimand is the realised funding fraction for this year's applications, not the mean award among funded applicants.
Final Answer
Not identifiable from the success-only extract. The fraction is \(18/N\), where total submitted applications \(N\) is unknown; reporting N (or the number of rejections) is sufficient.