This Exam P sample reference tests Variance and Standard Deviation. Independence makes the variance of the annual total equal to 52 times the common weekly variance. Equating this to the annual variance 9 gives weekly variance 9/52 and weekly standard deviation 3 divided by the square root of 52, so choice D is correct.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis divides a standard deviation by 52 squared. Variances add across independent weeks, so only one factor of 52 belongs in the variance equation.
BThis divides the annual standard deviation directly by 52, as though standard deviations rather than variances added for independent amounts.
CThis treats the annual standard deviation 3 as though it were the annual variance, producing the square root of 3/52 instead of 9/52.
EThis scales the annual standard deviation upward by the square root of 52; the weekly component must be smaller than the annual total.
Original practice · fully worked
Original variant: weighted navigation errors
Three independent navigation errors A, B, and C have standard deviations 2, 3, and 6 meters, respectively. A corrected position error is defined as T = 2A − B + 0.5C. Calculate the standard deviation of T.
A 5.000
B 5.831
C 5.916
D 7.000
E 10.000
Variant answer in brief
The three transformed errors have standard deviations 4, 3, and 3. Independence makes their variances add, so Var(T)=16+9+9=34 and SD(T)=√(34)=5.831, which is choice B.
Setup
Setup
Record the standard deviations and the coefficients in the corrected error.
SD(A)=2,SD(B)=3,SD(C)=6
T=2A−B+0.5C
Model
Model
For independent inputs, each variance is multiplied by the square of its coefficient before the contributions are added.
Var(T)=22Var(A)+(−1)2Var(B)+(0.5)2Var(C)
Compute
Compute
Insert the three component variances and take the square root of the resulting total variance.
Var(T)=4(22)+1(32)+0.25(62)=34
SD(T)=34=5.830951895…
Answer
Answer
The corrected position error has standard deviation approximately 5.831 meters.
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