Download Statistical Reasoning in Sports Worksheet#8.8 (p.282-284)

Statistical Reasoning in Sports Worksheet#8.8 (p.282-284)
Name _______________________ Per._____ Date _____________
My SAT score was at the 90th percentile. What does that mean?
The SAT (Scholastic Aptitude Test) is one of many standardized tests that are graded by determining
the mean score and standard deviation, then assigning grades according to how many standard deviations
above or below the mean an individual’s raw score is. The database of test scores is approximately
symmetric, unimodal, and bell-shaped, so a Normal Curve is used to approximate the distribution of test
scores. The score that is reported to you is usually a PERCENTILE value. To be at the 90 th percentile
means your score was about 1.28 standard deviations above the mean, because in the Normal Table,
.9000 is almost exactly equal to a z-score of 1.28.
The Normal Distribution is a distribution of data values that is perfectly symmetric, unimodal, and bellshaped. We can use the characteristics of a Normal Distribution to model and make predictions about
an athlete’s or team’s PERFORMANCES when the distribution of the PERFORMANCES is approximately
symmetric, unimodal, and bell-shaped. The Normal Distribution is the source of the 68-95-99.7 rule.
The graph of a Normal Distribution is called a Normal Curve. Locate and connect the points whose
coordinates are given; this will create a Normal Curve that is perfectly symmetric, unimodal, and bellshaped: (-3, 0.01) (-2.5, 0.02) (-2, 0.05) (-1.5, 0.13) (-1, 0.24) (-0.5, 0.35) (0, 0.40) (+0.5,
0.35) (+1, 0.24) (+1.5, 0.13) (+2, 0.05) (+2.5, 0.02) (+3, 0.01)
Fraction of data values
0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -
----------------------------------------------------------------------------------------------|
|
-3
|
|
-2
|
|
-1
|
|
0
|
|
+1
|
|
+2
|
|
+3
|
z-score (number of standard deviations above or below the mean)
DO NOT draw vertical lines at each whole number z-score. Instead, use the Normal Table and look
up the decimal equivalent of every 10th percentile ( .1000, .2000, etc.). Locate the closest z-score of
each of those and draw a vertical line on the Normal Curve at each z-score equivalent of those
percentiles. Also record them here:
Percentile 
Table value 
z-score 
0
10
20
30
40
50
60
70
80
90
100
( 1 ) Suppose that a basketball player’s scoring average places her at the 80 th percentile of all players in
her league. Explain what it means to be at the 80th percentile.
( 2 ) In the 2008 Wimbledon tennis tournament, Rafael Nadal averaged 115 miles per hour (mph) on his
first-serves. Assuming that the distribution of his first-serve speeds can be modeled by a Normal
Distribution with a standard deviation of 6 mph, use z-scores and the Normal Table to answer these
questions:
( a ) The fastest 15% of Nadal’s first-serves are above what speed?
( b ) What is the 25th percentile of his distribution of first-serve speeds? What is another name for
this boundary?
( 3 ) According to the 68 – 95 – 99.7 rule, about 68% of the data values in a Normal Distribution will be
within 1 standard deviation of the mean. Then it follows that 34% of the data values will be from z=-1
to z=0, and 34% of them will be from z=0 to z=+1. What percentile values are included in those 68%
that are within 1 standard deviation of the mean?
( 4 ) If you take a standardized test that has a normal distribution of scores, what would your
percentile score be if your test score was exactly equal to the mean of all scores?
( 5 ) Your friend is bragging that their score was at the 99th percentile. How many standard deviations
above the mean is the 99th percentile?