Download Excursions in Modern Mathematics Sixth Edition

Excursions in Modern
Mathematics
Sixth Edition
Peter Tannenbaum
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Chapter 16
Normal Distributions
Everything is Back
to Normal (Almost)
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Normal Distributions
Outline/learning Objectives
To identify and describe an approximately
normal distribution.
 To state properties of a normal
distribution.
 To understand a data set in terms of
standardized data values.
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Normal Distributions
Outline/learning Objectives
To state the 68-95-99.7 rule.
 To apply the honest and dishonest-coin
principles to understand the concept of a
confidence interval.
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Normal Distributions
16.1 Approximately Normal
Distributions of Data
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Normal Distributions
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Approximately normal distribution
Data sets that can be described as having bar
graphs that roughly fit a bell-shaped pattern.
Normal distribution
A distribution of data that has a perfect bell
shape.
Normal curves
Perfect bell-shaped curves.
Normal Distributions
16.2 Normal Curves and
Normal Distributions
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Normal Distributions
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Symmetry
Every normal curve has a vertical axis of
symmetry, splitting the bell-shaped region
outlined by the curve into two identical halves.
We can refer to it as the line of symmetry.
Normal Distributions
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Median/mean.
We call the point of
intersection of the horizontal
axis and the line of symmetry
of the curve the center of the
distribution. The center is
both the median and the
mean (average) of the data.
We use the Greek letter 
(mu) to denote this value.
Normal Distributions
Median and Mean of a Normal Distribution
In a normal distribution, M =  . (If the
distribution is approximately normal, then M 
).
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The fact that the median equals the mean implies
that 50% of the data are less than or equal to the
mean and 50% of the data are greater than or
equal to the mean. For data fitting an
approximately normal distribution, the median
and the mean should be close to each other but
Normal Distributions
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Standard deviation.
The easiest way to describe the standard
deviation of a normal distribution is to look at
the normal curve. If you bend a piece of wire
into a bell-shaped normal curve at the very top,
you would be bending the wire downward (a),
Normal Distributions
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Standard deviation.
but at the bottom you would be bending the
wire upward (b). As you move your hands
down the wire, the curvature gradually
changes, and there is one point on each side of
Normal Distributions
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Standard deviation.
the curve where the transition from being bent
downward to being bent upward takes place.
Such a point P (in figure c) is called a point of
inflection of the curve.
Normal Distributions
The standard deviation of a
normal distribution is the
horizontal distance between
the line of symmetry of the
curve and one of the two
points of inflection (P or P' )
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Normal Distributions
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Standard Deviation of a Normal Distribution
In a normal distribution, the standard deviation
 equals the distance between a point of
inflection and the line of symmetry of the curve.
Quartiles of a Normal Distribution
In a normal distribution, Q3   + (0.675)  and
Q1   + (0.675) .
Normal Distributions
16.3 Standardizing
Normal Data
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Normal Distributions
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Standardizing
To standardize a data value x, we measure
how far x has strayed from the mean  using
the standard deviation  as the unit of
measurement.
Z-value
A standardized data value.
Normal Distributions
Standardizing Rule
In a normal distribution with mean  and
standard deviation  , the standardized
value of a data point x is z = (x - )/ .
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Normal Distributions
From x to z: Part 2
A normal distributed data set with mean  =
63.18lb and standard deviation  = 13.27lb.
What is the standardized value of x = 91.54lb?
z = (91.54 – 63.18)/13.27 = 2.13715…  2.14
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Normal Distributions
16.4 The 68-9599.7 Rule
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Normal Distributions
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The 68-95-99.7 Rule
1. In every normal distribution, about 68% of all the data
values fall within one standard deviation above and
below the mean. In other words, 68% of all the data
have standardized values between z = -1 and z = 1.
Normal Distributions
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The 68-95-99.7 Rule
1 (cont). The remaining 32% are divided equally
between data with standardized values z  -1 and data
with standardized values z  1 (see figure a).
Normal Distributions
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The 68-95-99.7 Rule
2. In every normal distribution, about 95% of all the data
values fall within two standard deviations above and
below the mean. In other words, 95% of all the data
have standardized values between z = -2 and z = 2.
Normal Distributions
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The 68-95-99.7 Rule
2 (cont). The remaining 5% of the data are divided
equally between data with standardized values z  -2
and data with standardized values z  2. (see figure b).
Normal Distributions
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The 68-95-99.7 Rule
3. In every normal distribution, about 99.7% (practically
100%) of all the data values fall within three standard
deviations above and below the mean. In other words,
99.7% of all the data have standardized values between z
= -3 and z = 3. There is a minuscule amount of data with
standardized values outside the range (see figure b).
Normal Distributions
16.5 Normal Curves
as Models of RealLife Data Sets
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Normal Distributions
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The 68-95-99.7 Rule for Normal curves
1. About 68% of the data values fall within (plus
or minus) one standard deviation of the mean.
2. About 95% of the data values fall within (plus
or minus) two standard deviations of the
mean.
3. About 99.7%, or practically 100%, of the data
values fall within (plus or minus) three
standard deviations of the mean.
Normal Distributions
16.6 Distributions of
Random Events
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Normal Distributions
Coin-Tossing
Experiments: Part 1
Distribution of random
variable X (number of
Heads in 100 coin
tosses) (a) 10 times,
(b) 100 times, (c) 500
times, (d) 1000 times,
(e) 5000 times, and
(f) 10,000 times.
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Normal Distributions
16.7 Statistical Inference
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Normal Distributions
The Honest-Coin Principle
Suppose an honest coin is tossed n times (n 
30), and let X denote the number of Heads that
come up. The random variable X has an
approximately normal distribution with mean
 = n/2 Heads and standard deviation
heads   ( n ) / 2
.
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Normal Distributions
Coin-Tossing Experiments: Part 2
An honest coin is going to be tossed 256 times.
Let’s say that we can make a bet that if the
number of Heads tossed falls somewhere
between 120 and 136, we will win; otherwise
we will lose. Should we make such a bet?
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Normal Distributions
Coin-Tossing Experiments: Part 2
X = 256
 = n/2 = 256/2 = 128
  ( 256) / 2  8
The values 120 to 136 are exactly one standard
deviation below and above the mean of 128, which
means that there is a 68% chance that the number of
Heads will fall somewhere between 120 and 136.
We should indeed make this bet!
  ( n) / 2
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Normal Distributions
The Dishonest-Coin Principle
Suppose an arbitrary coin is tossed n times (n  30),
and let X denote the number of Heads that come up.
Suppose also that p is the probability of the coin
landing heads, and (1 – p) is the probability of the coin
landing tails. Then the random variable X has an
approximately normal distribution with mean  = n • p
Heads and standard deviation   n  p  1  p  Heads.
.
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Normal Distributions
Coin-Tossing Experiments: Part 3
Let p = 0.20 n = 100
What can we say about X?
 = n • p = 100  0.20 = 20
  n  p  1  p 
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=4
Normal Distributions
Coin-Tossing Experiments: Part 3
Applying the 68-95-99.7 rule gives the following:
 There is about a 68% chance that X will be
somewhere between 16 and 24.
 There is about a 95% chance that X will be
somewhere between 12 and 28.
 The number of Heads is almost guaranteed (about
99.7% chance) to fall somewhere between 8 and 32.
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Normal Distributions
Conclusion
 Bell-shaped
(normal) curves
 Statistical Inference
 Laws of probability
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