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### 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?

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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