This Exam P sample reference tests Normal Distribution. The probability condition gives a mean-to-standard-deviation ratio of Phi inverse 0.92, or 1.405072. Combining this ratio with E[X²]=mu²+sigma²=74 yields variance 24.8804, so choice A is correct.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
BThis uses mu squared equal to z times sigma squared instead of z squared times sigma squared, producing 74/(1+1.40507)=30.77.
CThis writes the second-moment identity as mu squared plus sigma rather than mu squared plus sigma squared, leading to a reported variance near 34.50.
DThis corresponds to using an incorrect standardized distance of about 0.82 in place of the 92nd-percentile value 1.40507.
EThis writes the second moment as mu plus sigma squared, omitting the square on the mean and producing variance about 62.86.
Original practice · fully worked
Original variant: centered gauge readings
A precision gauge reading X is normally distributed with mean 50. The probability that X lies between 44 and 56 is 0.6826894921. Calculate the second raw moment E[X²].
A 36
B 2500
C 2506
D 2536
E 3136
Variant answer in brief
The interval is centered at the mean and contains the standard-normal probability between -1 and 1, so the gauge standard deviation is 6. The second raw moment is 50 squared plus 6 squared, or 2536, which is choice D.
Setup
Setup
Standardize the two endpoints around the known mean.
Pr(44<X<56)=Pr(−σ6<Z<σ6)
Model
Model
The stated central probability equals the standard-normal probability between minus one and one.
Φ(1)−Φ(−1)=0.6826894921…
σ6=1
Compute
Compute
Recover the standard deviation and apply the second-moment identity.
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