The Scholarship Score Cutoff
A regional scholarship awards $8,000 to students whose composite score exceeds a cutoff. Composite = 0.6 × GPA-points + 0.4 × test-percentile, where GPA-points range 0–100 and test-percentile ranges 0–100. Last year 240 students applied and 60 won, with a cutoff of 78. This year 300 students apply. The committee notices the applicant pool's mean composite rose from 71 to 74, and the standard deviation stayed near 9. A board member says: 'More applicants and a higher average mean more winners, so we should raise the cutoff to keep costs at $480,000.' The budget is fixed at $480,000, meaning exactly 60 awards.
1. Using the normal approximation with mean 74 and standard deviation 9, estimate how many of the 300 applicants would score above 78. Compare with the 60 awards available and advise the committee.
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Standardize the cutoff: z = (78 - 74) / 9 = 4/9 ≈ 0.444.
Tail probability: P(Z > 0.444) ≈ 0.328 (using a standard normal table; between z = 0.44 and 0.45).
Expected count: 0.328 × 300 ≈ 98 applicants above 78.
Budget check: At $8,000 each, 98 awards cost $784,000 - far over the $480,000 cap of 60 awards.
Advice: The board member's logic is flawed. A higher mean with the same spread pushes more people over any fixed threshold, not fewer, so raising the cutoff does not 'keep costs at $480,000' unless the new cutoff is chosen to yield exactly 60 winners. To find that cutoff, solve for the 80th percentile of the composite distribution (since 60/300 = 0.20 must be above it): z ≈ 0.842, so cutoff = 74 + 0.842 × 9 ≈ 81.6. The committee should set the cutoff near 82, not simply 'raise it,' and should note the normal approximation is rough for a bounded 0–100 score.
2. A committee member proposes weighting GPA at 0.8 and test-percentile at 0.2 to 'reward sustained performance.' Explain one statistical consequence of this change for the cutoff calculation, assuming GPA and test scores are positively correlated.
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Changing the weights changes both the mean and the standard deviation of the composite, so the old cutoff of 78 - and even the recomputed 81.6 - no longer applies directly.
Mean: If GPA-points have mean μ_G and test-percentile mean μ_T, the new mean is 0.8μ_G + 0.2μ_T. Because GPA is typically higher and less variable than test percentiles in many pools, the mean may shift.
Variance: For correlated variables, Var(0.8G + 0.2T) = 0.64σ_G² + 0.04σ_T² + 2(0.8)(0.2)ρσ_Gσ_T. Positive correlation (ρ > 0) inflates the composite variance relative to the uncorrelated case, because the covariance term is positive. A larger standard deviation spreads the distribution, so for a fixed 20% award rate the cutoff sits further above the mean in absolute points.
Practical consequence: The committee cannot reuse prior cutoffs; it must recompute the mean and standard deviation of the new composite from the applicant data, then take the 80th percentile. Also, weighting GPA more heavily may compress scores if GPA has a lower standard deviation than test scores, which would lower the spread and pull the cutoff closer to the mean.