Enrollment Funnel With an Interview Cap
A specialized master's program receives 1,000 applications each year. The admissions funnel has three stages: 30% of applicants are shortlisted for interview; 50% of interviewees are admitted; and 40% of admitted students enroll. The faculty can conduct at most 360 interviews in total. The program director has a target of 75 enrolled students. She proposes raising the interview shortlist rate from 30% to 36%, arguing that 'a 6% increase in interviews will close the gap.' A colleague replies that the proposal may hit the interview cap and still miss the target.
The program cannot change the admit rate or the yield in the current cycle, but it can change the shortlist rate up to the cap. All percentages are applied to the relevant stage population and rounded only for reporting.
1. Compute the current number of enrolled students. Then compute the enrollment under the director's proposed 36% shortlist rate.
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Current funnel:
- Interviews: 1,000 × 0.30 = 300
- Admits: 300 × 0.50 = 150
- Enrolled: 150 × 0.40 = 60
Proposed 36% shortlist:
- Interviews: 1,000 × 0.36 = 360, which exactly equals the interview cap
- Admits: 360 × 0.50 = 180
- Enrolled: 180 × 0.40 = 72
The proposal raises enrollment from 60 to 72, leaving a gap of 3 students against the target of 75.
2. Evaluate the director's claim that 'a 6% increase in interviews will close the gap.' Address both the arithmetic and the terminology.
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The claim is wrong on two counts. First, the arithmetic: the proposal produces 72 enrolled students, not 75, so it does not close the gap. It also consumes the entire interview cap of 360, leaving no room to increase the shortlist further without exceeding faculty capacity.
Second, the terminology: moving from 30% to 36% is an increase of 6 percentage points, not a 6% increase. The relative increase in interviews is (360 − 300) / 300 = 60 / 300 = 20%. A '6% increase' would mean 30% × 1.06 = 31.8%, which would produce only 318 interviews and 63.6 enrolled students, far from the target. The distinction matters because percentage-point changes and percent changes are different quantities.
3. With the shortlist at the cap of 360 interviews and the admit rate fixed at 50%, what yield would be required to reach 75 enrolled students? Interpret the result.
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At the cap:
- Interviews = 360
- Admits = 360 × 0.50 = 180
Required yield y satisfies 180 × y = 75, so y = 75 / 180 = 0.41666..., or about 41.7%.
The program would need to raise yield from 40% to roughly 41.7%, an increase of about 1.7 percentage points. That is a modest but real change: it is not enough to increase the shortlist rate alone, because the interview cap binds. Alternatively, if yield remained at 40%, the program would need an admit rate of 75 / (360 × 0.40) = 75 / 144 ≈ 52.1%, or about 2.1 percentage points above 50%. The key judgement is that the target is feasible only if at least one downstream conversion rate improves; the shortlist lever alone cannot reach 75 under the cap.