Statistics 21: Homework 3 Solutions
1. A student scores at the 90th percentile on a midterm examination. All previous semesters of
the class have resulted in an approximately football shaped scatter plot of final exam scores
versus midterm scores, and the historical correlation between these two quantities is consis
tently 0.8. Predict the student’s percentile on the final exam.
We can answer this question in 3 steps: first, we will determine the student’s midterm score in
standard units using the Normal distribution; second, we will predict the student’s final exam
score in standard units using the regression equation; third, we will convert this predicted
score to a percentile using the Normal distribution.
Step 1: Although we don’t know the student’s score, the footballshaped scatter plot ensures
that the midterm scores follow a Normal distribution.
We can therefore find the student’s
score in standard units by finding the value of
z
that corresponds to the 90th percentile in
the standard Normal distribution.
Because 90% of the data are less than
z
, then 10% of
the data are greater than
z
, so we have the right tail area. By the symmetry of the Normal
distribution, the left tail area is also 10%, and therefore the middle region has area 80%. We
can find
z
by locating the value in the table with an Area measurement closest to 80%; this is
given by
z
= 1
.
3 with area 80
.
64%. Therefore, the student scored
z
= 1
.
3 standard deviations
above the mean. We will call this value
M
= 1
.
3 to represent the midterm score in standard
units;
M
∼
N
(0
,
1).
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 Spring '08
 anderes
 Statistics, Normal Distribution, Standard Deviation

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