Original practice · fully worked
Original variant: equipment-service reimbursements
A service program models each equipment repair cost as normal with mean 850 dollars and standard deviation 120 dollars. A deductible d applies to each repair. Only 8% of repairs generate a reimbursement greater than 300 dollars. Calculate d.
- A
169 - B
550 - C
681 - D
719 - E
1019
Answer and calculation
A reimbursement above 300 requires a repair cost above d+300. The 92nd percentile of the repair-cost distribution is 1018.609, leaving a deductible of 718.609 and choice D.
Setup
Setup
Represent the repair cost by X and the reimbursement by the positive part after the deductible.
X∼N(850,1202),R=(X−d)+ Model
Model
Translate the reimbursement event into a repair-cost event and express the upper tail as a percentile.
Pr(R>300)=Pr(X>d+300)=0.08 Pr(X≤d+300)=0.92 Compute
Compute
Use the 92nd standard-normal percentile and isolate d.
Φ−1(0.92)=1.4050715603… d=850+120(1.4050715603…)−300=718.6085872… Answer
Answer
The deductible is approximately 719 dollars.
d≈719(D) 