Original practice · fully worked
Original variant: symmetric operating band for a ping count
During an extended warehouse scan, the number N of defect pings follows a Poisson law with expected count 3,600. Using a normal approximation, determine the smallest whole number h for which the symmetric band from 3,600-h through 3,600+h contains approximately 90% or more of the count distribution.
- A
60 - B
90 - C
98 - D
99 - E
197
Answer and calculation
The approximating normal distribution has mean 3,600 and standard deviation 60. Central 90% coverage requires the 95th standard-normal percentile, so h must be at least 1.644854(60)=98.6912; the smallest whole number is 99, choice D.
Setup
Setup
A Poisson variable has variance equal to its mean, which determines the scale of the approximating normal distribution.
E[N]=Var(N)=3600,SD(N)=60 Model
Model
A central 90% band leaves 5% in each tail, so its positive standardized endpoint is the 95th percentile.
Pr(−z≤Z≤z)=0.90,z=Φ−1(0.95)=1.644853627 Compute
Compute
Rescale the standard-normal half-width and round upward because h must be a whole number meeting the coverage target.
h≥60(1.644853627)=98.69121762 hmin=99 Answer
Answer
The narrowest eligible integer half-width is 99 pings.
99(D) 