This Exam P sample reference tests Normal Percentile Translation. The 1982 93rd-percentile score is about 647.6 and is reported as 650. Under the 2008 mean and standard deviation 101, this gives z=1.2772 and percentile 89.9%, choice B.
How to solve this Normal Percentile Translation question
Setup
Setup
Convert the 93rd percentile in the first score distribution to a numerical score using the corresponding standard-normal quantile.
z0.93=1.47579,503+98z0.93=647.63⟶650
Model
Model
The first-distribution score is approximately 647.6 and is reported as 650. Standardize that rounded score using the second distribution's mean and standard deviation.
znew=101650−521=1.27723
Compute
Compute
The resulting standard score is approximately 1.2772, whose cumulative probability is about 0.89924. This corresponds to the 90th percentile.
Φ(1.27723)=0.89924
Answer
Answer
The rounded score corresponds to approximately the 90th percentile.
90th percentile(B)
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AThe value 89th rounds the cumulative probability downward to the lower integer rather than to the nearest percentile.
EThe 93rd percentile simply carries the original percentile into the new distribution without restandardizing the rounded score.
Original practice · fully worked
Original variant: translating a rounded score across years
In year 1, scores are normal with mean 600 and standard deviation 80. A candidate at the 90th percentile has the score rounded to the nearest ten. In year 2, scores have mean 620 and standard deviation 75. Find the percentile of that same rounded score in year 2.
A 84th
B 85th
C 86th
D 87th
E 90th
Variant answer in brief
The year-1 90th-percentile score is about 702.5 and rounds to 700. In year 2 this is z=1.0667, whose CDF is about 0.857, choice C.
Setup
Setup
Convert the year-one 90th percentile to a score using the standard-normal 90th-percentile quantile.
600+80(1.28155)=702.52⟶700
Model
Model
The score is approximately 702.5 and rounds to 700. Standardize 700 under the year-two distribution.
zyear2=75700−620=1.06667
Compute
Compute
The year-two standard score is approximately 1.0667, giving cumulative probability about 0.857 and hence the 86th percentile.
Φ(1.06667)≈0.857
Answer
Answer
The same rounded score is approximately at the 86th percentile.
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