This Exam P sample reference tests Poisson Distribution. This is a normal approximation to an aggregate Poisson count. The 1,600 independent daily counts combine to mean and variance 6,400, so the stated boundary has z=1.2 and upper-tail probability 0.1151, selecting choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 0.01 is close to the upper tail at z=2.4. That z-score follows from using √(1600)=40 as the aggregate standard deviation and omitting the daily Poisson variance factor 4.
CThe value 0.19 is the upper tail near z=0.88, which would require an aggregate standard deviation near 109 instead of the verified value 80.
DThe value 0.27 is close to the upper tail at z=0.60. This results from halving the 96-count excess before dividing by the correct standard deviation 80.
EThe value 0.49 results from dividing the 96-count excess by the variance 6,400, producing z=0.015. Standardization divides by the standard deviation 80.
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
Variant answer in brief
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.
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