Download 1 The Normal Curve and the 68 95 99.7 Rule

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The Normal Curve and the 68-95-99.7 Rule
The normal curve (also known as the bell curve) is the most common distribution of data. The normal curve is completely determined by two parameters:
mean and standard deviation. The normal curve is symmetric about the mean
which is also the median and the mode. Most data is clumped in close to the
mean.
Figure 1: A very plain normal curve
The normal distribution is important since it tells us the amount of data that
falls in particular intervals relative to the mean and standard deviation. The
notation N ( ; ) states that the data has a normal distribution with mean and
a standard deviation . We use the notation mean and standard deviation
to indicate that these are de…ning parameters for a statistical distribution rather
than statistical values we computed from a sample. For example, N (35; 2:3)
indicates a normal distribution with a mean of 35 and a standard deviation of
2.3.
Theorem 1 The 68-95-99.7 Rule: In every normal distribution with mean
and standard deviation , approximately 68% of the data falls within one
standard deviation of the mean. Approximately 95% of the data falls within
two standard deviations of the mean. And …nally, approximately 99.7% (almost everything) of the data falls within three standard deviations of the mean.
This rule is illustrated in Figure 2 from http://www.answers.com/topic/normaldistribution.
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In the normal distribution all measurements are computed in terms of distance from the mean relative to standard deviation. Think of standard deviation as a ruler.
Professor Russell and Professor Johnson teach di¤erent sections of Elementary Statistics. On test 1, Professor Russell’s class had an average score of 10
points with a standard deviation of 2 points. Assume these test grades follow
a normal distribution and explain.
Problem 1 What grades comprise the central 68% of the students in Professor
Russell’s class?
Problem 2 What grades comprise the central 99.7% of the students in Professor Johnson’s class?
Remark 2 A useful tool to convert to standardized units is the z-score
z=
x
:
Proper use of a z-score permits the comparison of apples to oranges!
Problem 3 On test 1, Professor Johnson’s class had an average score of 150
points with a standard deviation of 10 points. Whose score is better: Mary in
Russell’s class with a score of 14 or Dawn in Johnson’s class with a score of
160?
Problem 4 In Professor Russell’s class, Alan’s test score is z = 1:5. What is
Alan’s original test score?
Problem 5 In Professor Johnson’s class, what percentage of students scored
between 140 and 160 points?
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Problem 6 In Professor Johnson’s class, what percentage of students scored
more than 160 points?
Problem 7 If Professor Johnson’s class has 200 students, approximately how
many scored between 150 and 170 points?
At a particular high school, the average time to run a mile on the track team
is 6.7 minutes with a standard deviation of 1.02 minutes. On the archery team
the average number of bulls-eyes (out of 20 attempts) is 7.8 with a standard
deviation of 3.3.
Problem 8 David runs track while Jenny is an archer. If David runs a mile in
5 minutes and Jenny hits 12 bulls-eyes, whose performance is more outstanding?
Assume this data follows a normal distribution.
Problem 9 Is the sign on the z-score important? Explain.
Problem 10 The number of bulls-eyes made by Jack has a z-score of z =
1:15: How many times did Jack hit the bulls-eye?
De…nition 3 An observation that is unusually large or small, relative to the
other values in a data set, is called an outlier. An observation is considered to
be unusually large or small if the absolute value of its z-score is at least 2.
Problem 11 Is it unusual to be on the track team and run a 10 minute mile?
Problem 12 Is the score 4 unusual on test 1 in Professor Russell’s class?
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Problem 13 What is the minimum number of bulls-eyes one can hit and have
an unusually good performance?
The 68-95-99.7 rule is a nice beginning from which to explore the normal
curve. It would be greatly limiting if we could only work with z-scores of
1; 2 and 3. Fortunately, technology allows us to work with any z-score in
any normal distribution.
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Exercises
1. The average score in a game of Scrabble is 141 points with a standard
deviation of 7 points.
i. what scores represent the central 68% of all Scrabble scores?
ii. what scores represent the central 95% of all Scrabble scores?
iii. what percentage of scores fall between 148 and 155?
iv. is 160 an unusual score?
v. Tony has a z-score of z = 2:57 representing the last Scrabble game
he played. What is Tony’s raw score?
2. When opponents of the Durham Bulls baseball team face pitcher, Ebby
Calvin "Nuke" LaLoosh, they average 15 hits per game with a standard
deviation of 3 hits and this data follows a normal distribution.
i. In a particular game, the Winston-Salem Spirits get 10 hits against
Nuke. Find the corresponding z-score.
ii. Find the percentage of games where opponents get more than 21 hits.
iii. Is unusual for Nuke to throw a no-hitter1 ? Carefully explain your
answer and include all supporting computations..
3. Find all salary outliers for the ’97-’98 Chicago Bulls.
standard deviation of these salaries is $8,182,474.38.
1 In
Note that the
baseball, a pitcher throws a no-hitter if the opposing batters never get a hit.
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Player
Salary
1 Michael Jordan $33,140,000
2 Ron Harper
$4,560,000
3 Toni Kukoc
$4,560,000
4 Dennis Rodman $4,500,000
5 Luc Longley
$3,184,900
6 Scottie Pippen
$2,775,000
7 Bill Wennington $1,800,000
8 Scott Burrell
$1,430,000
9 Randy Brown
$1,260,000
10 Robert Parish
$1,150,000
11 Jason Caffey
$850,920
12 Steve Kerr
$750,000
13 Keith Booth
$597,600
14 Jud Buechler
$500,000
15 Joe Kleine
$272,250
Average
$4,088,711
Median
$1,430,000
Chicago Bulls
Salaries 1997-1998
Season
4. Find all salary outliers for the ’10-’11 Atlanta Hawks.
Atlanta Hawks Salaries
2010-2011 Season
Clark: Read section 2.1. Do problem 6.
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