Verified solution · Exam P
This Exam P sample reference tests Normal Distribution. The variance factor 2.25 corresponds to a standard-deviation factor 1.5. A common value therefore has standardized score z_0.14/1.5 under the wider distribution, giving percentile 23.57%, choice D.
Independent solution
Setup
Translate the variance relationship into the scale relationship used by normal quantiles.
Model
The two distributions share a mean, so equate the two representations of the common percentile value.
Compute
Evaluate the lower-tail score and map the rescaled score back through the normal CDF.
Answer
The matching percentile for the wider distribution is approximately 23.6.
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Original practice · fully worked
Two automated lines produce measurement errors centered at a shared value m. Each error law is Gaussian, but the spreads may differ. At one fixed trigger h greater than m, line A crosses h on 2.5% of runs, whereas line B crosses h on 10% of runs. Find the quotient of B's variance and A's variance.
The common alarm distance equals 1.95996 standard deviations for A and 1.28155 for B. Their standard-deviation ratio is 1.52937, so the variance ratio is 2.33897, choice E.
Setup
Let h be the shared alarm level and convert each upper-tail probability to a positive z-score.
Model
Both standardized distances represent the same physical distance h-mu.
Compute
Find the scale ratio and square it to obtain the requested variance ratio.
Answer
Process B has about 2.339 times the variance of process A.
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ActuaryProof. “SOA Exam P Sample Question #165 — Verified Solution.” Verified 2026-07-31.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
