Problems: Z-Scores
1. The average IQ of a group of people is 105 with a
standard deviation of 15. What is the standardized (or z)
score of:
- a) someone with an IQ of 93?
b) someone with an IQ of 135?
2. Suppose a tree farm finds that the mean height of
three-year-old trees is 47 inches with a standard deviation
of 3 inches.
- a) Compute the z-score of a tree 51 inches tall.
b) Compute the z-score of a tree 49 inches tall.
c) A tree has a z-score of -3. How tall is it?
3. On the standard IQ test, the mean is 100 and the
standard deviation is 15.
- a) Compute the z-score of a person with an IQ of
120.
b) Compute the z-score of person with an IQ of 20.
c) A person has a z-score of +3. What is her IQ?
4. A group of people is tested to see how much weight
they can lift. The mean lift is 120 pounds with a standard
deviation of 20 pounds. Compute the z-score for someone who
lifts
- a) 125 pounds.
b) 80 pounds.
c) If someone has a z-score of +3, how much can that
person lift?
5. In the not too distant past, a professor at some
prestigious school (it might have been Duke) suggested that
in an effort to combat grade inflation, the school should
issue standardized grades (or z-scores) in addition to the
regular grades.
Let us see how this might work. Suppose Sue was in three
classes, each with 20 students. In Introductory Alchemy, the
professor awarded an A to 15 students and an A- to 5. The
average grade point in this class was 3.90 and the standard
deviation was .15. In her Intermediate Astrology class, the
professor was a little tougher, giving five As, five Bs,
five A-s, and five B+s. The GPA of the class was 3.50 with a
standard deviation of .38. In her class of Ancient
Phoenician literature, the prof gave mostly Bs, with a
smattering of other grades, for a GPA of 2.72 and a standard
deviation of .7
Sue got an A- (3.67) in Alchemy, a B+ (3.33) in
Astrology, and a B (3.00) in Phoenician lit. Compute her
standardized grade in each class. If we judged by
standardized grades, where did she do best? Where did she do
worst? Explain carefully.
6. A test in economics has an average grade of 12 with a
standard deviation of 3. The high score on the quiz was 20.
How many standard deviations above the mean was this
student?
7. A group of students has an average height of 6 feet
with a standard deviation of .25 feet.
- a) Suppose we had measured in inches instead. Can we
tell what their mean and standard deviation would be? If
we can tell, what would they be?
- b) A student is 6.4 feet tall. How many standard
deviations away from the average is he?
- c) How tall is a student who is 2 standard deviations
below the average?
-
Answers here
More problems:
8. A group of people is tested to see how much weight
they can lift. The mean lift is 120 pounds with a standard
deviation of 20 pounds. Compute the z-scores for someone who
lifts 120 pounds with a standard deviation of 20 pounds.
Compute the z-scores for someone who lifts:
- a) 150 pounds.
- b) 90 pounds.
- c) If someone has a z-score of -2, how much can he or
she lift?
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