This Exam P sample reference tests Discrete Distributions. The mean is 55 and the standard deviation is 21.79. Values 40, 50, 60, and 70 lie within one standard deviation and carry total probability 45%, choice A.
Compute the mean and second moment from the discrete probability table, then use them to determine the interval within one standard deviation of the mean.
E[X]=x∑xpx
E[X2]=x∑x2px
Model
Model
The table gives first moment 55 and second raw moment 3500, so the standard deviation is the square root of 3500 minus 55 squared.
E[X]=55
E[X2]=3500
σ=3500−552=21.7945
Compute
Compute
The one-standard-deviation interval is [33.2055,76.7945]. Exactly the support values 40, 50, 60, and 70 lie inside it, with total mass 0.45.
55−21.7945≤X≤55+21.7945
X∈{40,50,60,70}
0.05+0.20+0.10+0.10=0.45
Answer
Answer
Therefore 45% of the distribution lies within one standard deviation, corresponding to choice A.
45%(A)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
BThe value 55% includes one additional support point whose distance from 55 exceeds 21.7945.
CThe value 68% imports the normal-distribution rule of thumb. The supplied distribution is discrete, so its listed masses must be summed directly.
DThe value 85% includes both neighboring extreme support points without checking them against [33.2055,76.7945].
EThe value 100% assumes every support point lies in the interval, contrary to the explicit endpoint comparison.
Original practice · fully worked
Original variant: scores inside one empirical standard deviation
A quality score takes values 1, 3, 5, 7, and 9 with probabilities 0.10, 0.20, 0.40, 0.20, and 0.10. What probability lies within one standard deviation of its mean, including endpoints?
A 0.20
B 0.40
C 0.60
D 0.80
E 1.00
Variant answer in brief
Symmetry gives mean 5 and variance 4.8, so the interval is about [2.81,7.19]. Scores 3, 5, and 7 contribute probability 0.80.
Setup
Setup
Use symmetry of the five-point score distribution to obtain mean 5, then compute its variance from squared deviations about 5.
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