Chapter 7: The Normal Probability Distribution

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Chapter 7 The Normal Probability Distribution
7.1 Properties of the Normal Distribution 7.2 The
Standard Normal Distribution 7.3 Applications of
the Normal Distribution 7.4 Assessing
Normality 7.5 The Normal Approximation to the
Binomial Probability Distribution
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December 8, 2008
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Properties of the Normal Distribution

• In this chapter we study a probability
  distribution for a continuous random variable,
  called the Normal Distribution. This
  distribution is studied for several reasons
• It is a good model for the distribution of many
  different populations.
• (2) Several probability distributions (including
  some discrete probability distributions) can be
  approximated by a Normal Distribution.
• (3) It is bell-shaped and hence, the Empirical
  Rule applies.
• (4) Many inferential methods in statistics is
  based on the assumption that the population is
  distributed according to a Normal Distribution.
• Hence, it is ubiquitous. If you want to have
  detailed knowledge of only one probability
  distribution, then the Normal Distribution is one
  to study.

Section 7.1
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Continuous Random Variables

• A continuous random variable has a continuum of
  possible values.
• Examples time, age, height and weight.
• A continuous random variable has a continuous
  probability distribution that is a curve that is
  defined on the interval from which X takes its
  values.

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Probability Distribution of a Continuous Random
Variable
Definition Let X be a continuous random
variable. Suppose that values of X, i.e., x,
lie in an interval a,b. The probability
distribution of X is a function, f(x), that is
define on a,b, such that the area under the
graph of f is equal to 1. The function, f(x), is
also called the probability density function
(PDF) of the distribution. Note It is possible
that either a and/or b are infinity.
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Probabilities and Continuous Probability
Distributions
In the discrete case, we can extend the
probability of x (say at x 2) to the interval
1.5,2.5. The probability for any x in
1.5,2.5 will be P(2). This probability is
equal to the area of the rectangle whose base is
the interval 1.5,2.5 and the height is P(2).
This manner we can extend a discrete probability
distribution to a continuous probability
distribution that is defined on an intervals.
For example, the probability for any x in
1.5,2.5 is P(2) which is area of the rectangle
constructed above.
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Area and Discrete Probability Distribution
Recall If x1 lt x2 lt lt xN, then P(x xk)
P(x1) P(x2) P(xk).
From the histogram of the discrete probability
distribution, the quantity, P(x1) P(x2)
P(xk), is related to the area of the bars in the
histogram. In fact, if the width of the bars are
1, then it is exactly the sum of the areas of the
bars from x1 to xk. Hence, P(x xk) is an area
under the bar.

• Note
• P(x xN) 1
• If m lt n, then P(xm x xn) is the sum of the
  areas of the bars from xm to xn.

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Probabilities and Continuous Probability
Distributions
For a continuous probability distribution, we
generalize the ideas presented for the discrete
probability distribution. Let us consider some
interval ?,? in the interval a,b. We want to
associate a probability for x in the interval
?,?. We define the probability for x in the
interval ?,? as the area under the curve of
f(x) and above the interval ?,? .
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Cumulative Probability Distribution
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Continuous-Discrete Probability Distribution of a
Random Variable
ExampleThe random variable is the height of
females in a certain population.
As the number of possible outcomes for a random
variable X becomes large, the discrete
probability distribution can approach a
continuous probability distribution.
We can often approximate discrete probability
distribution by continuous probability
distributions.
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Remark
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Mean and Standard Deviation of a Continuous
Probability Distribution
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Summary of a Probability Distribution for a
Continuous Random Variable
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The Uniform Probability Distribution
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Normal Probability Distribution
We now examine a particular probability
distribution for a continuous random variable
that takes all values of the real line.
Remark The function f(x) is called a probability
density function and is abbreviated as PDF. We
shall call the probability distribution, given by
the above probability distribution function, the
Normal Distribution.
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Remark
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Dependence on Mean and Standard Deviation
? 0 and ? 1
? 0 and ? 3
? 2 and ? 1
? -2 and ? 1
We will call the graph of f(x) the normal density
curve or simply, the normal curve.
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Computing the Probability Distribution Function
for the Normal Curve

• How can you calculate the function f(x) for
  different values of x? Once you have define ?
  and ?, you use
• calculator
• computer
• tables

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Facts about the Normal Distribution

• Here are some properties of the graph of the
  normal density function f(x)
• It is symmetric with respect to the line x ??
• The highest value of the curve occurs when x
  ?.
• It has two points of inflection x ? ?. A
  point of inflection is were a curve changes from
  being concave upward to concave downward or
  vice-versa.
• The area under the curve is 1.
• It highest value of f(x) (at x ?) changes with
  ?, but is always positive.
• For some standard deviations, ?, the values of
  f(x) may be larger than 1.0 and hence,
  probability density function at a point, x, is
  not necessarily the probability, P(x).

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Some Useful Facts about the Normal Distribution
Function
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Empirical Rule for the Normal Distribution

• For the normal distribution and its curve, we
  have the following empirical rules for
  bell-shaped distributions
• Approximately 68 of the area under the curve
  lies in the interval ?-?, ??.
• Approximately 95 of the area under the curve
  lies in the interval ?-2?, ?2?.
• Approximately 99.7 of the area under the curve
  lies in the interval ?-3?, ?3?.
• Recall The empirical rule for bell-shaped
  distributions.

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The Normal Cumulative Probability Distribution
Definition The Cumulative Probability
Distribution, P(x ?), is defined to be the area
under the Normal Probability Density Function for
x ?. The value of P(x ?) is always between 0
and 1.
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Fact about P(x ?)
Fact The Normal Cumulative Probability
Distribution (Normal CPD) of x gives the
probability that x ?. For example, if X
denotes the continuous random variable which is
the weight of an individual randomly chosen from
a population that obeys a normal distribution and
x is the numerical value for this random
variable, then P(x 180) is the probability that
this individual weighs at least 180 pounds.
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Cumulative Probability Distribution of an Interval
Another Fact The normal cumulative probability
distribution for an interval ?,? is the area
under the curve and above the interval P(?? x
?).
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Example

• Suppose the replacement time of a particular
  brand of refrigerator is normally distributed
  with mean ? 14 years and standard deviation ?
  2.5 years.
• Sketch a graph of the probability density
  function and the cumulative probability density
  function.
• (b) Shade the region in the graph of the
  probability density function that represents the
  probability that a randomly selected refrigerator
  will last at least 17 years.
• (c) What is the probability that it will last
  more than 17 years.
• (d) What is the probability that it will be
  replaced between 14 years and 16.5 years.

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Calculation of the Cumulative Probability
Distribution on the TI-83

• 2nd VARS (DISTR) key
• Select normalcdf( ENTER
• Complete entry e.g., normalcdf(-1.9,2.3,0.5,1.7)
  ENTER
• Answer 0.7761502183

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z - score
Recall We introduce the concept of the z-score
for an observation in a sample z
(observation - mean)/(standard deviation) or
letting observation x, mean ? and standard
deviation ?, we have z (x - ?)/?. For
example, when z 1, then x ? ?. When z
2, then x ? 2?. In general, the z-score is
a measure of how far is the observation (x) from
the mean.
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z-score and the Normal Distribution

• Between z -1 and z 1, the values of x lie in
  the interval ?-?,??. We know from the
  empirical rule, this is approximately 68 of the
  total area under the normal curve.
• Between z -2 and z 2, the values of x lie in
  the interval ?-2?,?2?. We know from the
  empirical rule, this is approximately 95 of the
  total area under the normal curve.
• Between z -3 and z 3, the values of x lie in
  the interval ?-3?,?3?. We know from the
  empirical rule, this is approximately 99.7 of
  the total area under the normal curve.
• Hence, P(?-??? x ??) is approximately 0.68,
  P(?-2??? x ?2?) is approximately 0.95, and
  P(?-3??? x ?3?) is approximately 0.997.

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Standard Normal Distribution
Definition The normal distribution with ? 0
and ? 1 is called the Standard Normal
Distribution.
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The Standard Random Variable
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Example
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The Standard Normal Distribution
We observed in the previous section that every
Normal Distribution with mean ? and standard
deviation ? can be converted to a Standard Normal
Distribution by the change of random variable z
(x - ?)/?.
Normal Distribution Standard
Normal Distribution
Section 7.2
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Computing Probabilities with the Standard Normal
Distribution
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Example
Example The time between release from prison and
conviction for another crime for individuals
under the age of 40 is normally distributed
(i.e., the probability of these events happen is
governed by a Normal Distribution) with a mean of
30 months and a standard deviation of 6 months.
Find the probability that an individual who has
been released from prison will be convicted of
another crime within 24 months. Solution We
want to calculate P(x 24) with ? 30 and ?
6. We can use the standard normal distribution
by introducing the z-score. z (x - 30)/6 or
when x 24, then z (24 - 30)/6 -1. Now P(z
-1) 0.1587. Hence, 15.87 of the prisoners
will return within 2 years. Below are the
probability density function (PDF) and the
cumulative probability distribution (CPD).
Notice that P(x lt 0) is approximately zero.
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Calculating P(a z b) from Tables
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Inverse Problem Given the value of P(z a),
find a
Suppose that we are given the value of P(z a)
i.e., the area under a Standard Normal curve and
we want to determine the value of a.

• Methods
• Tables
• Calculator - invNorm

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Inverse Problem Given the value of P(-a z
a), find a
Suppose that we are given the value of P(-a z
a) i.e., the area under a Standard Normal curve
and we want to determine the value of a.
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Inverse Problem Given the value of P(z gt a),
find a
Suppose that we are given the value of P(z gt a)
i.e., the area under a Standard Normal curve and
we want to determine the value of a.
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Applications of the Normal Distribution
One important application of the Normal
Distribution is the following. Suppose a
variable x in a population (e.g., the height of
individuals in Math 127A) is distributed
according to a Normal Distribution with mean ?
and standard deviation ?. If we consider X to be
a continuous random variable, then what is the
probability that any randomly selected individual
from the population will satisfy a x  b?
That is, what is P(a x b)? Remark We
sometimes substitute the word proportion for
probability. That is, what proportion of the
population will the random variable x lie in the
interval a,b?
Section 7.3
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Example
The Accreditation Council for Graduate Medical
Education found that average hours worked by
medical residents was 81.7 hours per week with a
standard deviation of 6.9 hours. Suppose that we
assume that the number of hours per week worked
by medical residents is distributed by a Normal
Distribution with ? 81.7 and ? 6.9. (a) What
is the probability that a medical resident will
work more than 80 hours per week? (b) What is the
probability that a randomly selected resident
will work between 60 and 80 hours per week?
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Example

• The Timken Company manufactures ball bearings
  with a mean diameter of 5 mm. Due to the
  manufacturing process there is some variation in
  the diameters of the ball bearings. It has been
  calculated that the distribution of diameters is
  normally distributed with a mean of 5 and a
  standard deviation of 0.02 mm.
• (a) What proportion of the ball bearings have
  diameters which are greater than 5.03 mm?
• (b) Any ball bearing that is smaller than 4.95 mm
  in diameter or greater than 5.05 mm is discarded.
  What proportion of ball bearings is discarded?
• (c) In one day, 30,000 ball bearings are
  manufactured. How many would you expect to be
  discarded in a day?

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Assessing Normality
Suppose that a variable of a population X is
distributed according to an unknown distribution.
Is there a way that we can test if this unknown
distribution is actually a Normal
Distribution? One Approach Take a large finite
sample from the population and create a histogram
to see if the histogram has the characteristics
of a Normal Distribution i.e., it is bell-shaped.
However, being bell-shaped does not mean that it
is a Normal Distribution.
Section 7.4
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Another Approach
TI-83 NormProbPlot
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Example
Data 0.533226, 2.73637, 2.76095, 2.83428,
2.62008, 1.82784, 1.31128, 1.87577, 0.70117,
3.09077, 2.47481, 2.09632, 2.22858, 2.23172,
1.76795, 0.153967, 1.19405, 2.70018, 1.66897,
0.583992 Sorted Data 0.153967, 0.533226,
0.583992, 0.70117, 1.19405, 1.31128, 1.66897,
1.76795, 1.82784, 1.87577, 2.09632, 2.22858,
2.23172, 2.47481, 2.62008, 2.70018, 2.73637,
2.76095, 2.83428, 3.09077 Normal Scores
-1.86824, -1.40341, -1.12814, -0.919136,
-0.744143, -0.589456, -0.447768, -0.314572,
-0.186756, -0.0619316, 0.0619316, 0.186756,
0.314572, 0.447768, 0.589456, 0.744143, 0.919136,
1.12814, 1.40341, 1.86824 n 20
Note Data was generated by a Normal Distribution
with ? 2 and ? 0.75.
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Example
Data -8.21923, -2.74515, -0.386428, -0.677152,
4.02123, -0.826667, 9.17761, 6.45027, -2.31864,
6.53159, 7.68041, -1.54977, -0.988243, 3.35719,
5.98133, 4.44442, 4.03768, 9.3086, 6.4066,
-9.51397, -6.42983, 1.88659, -1.5584, 6.85724,
-8.2106, -5.36826, 8.82803, -2.46561, -2.23184,
5.45841 Sorted Data -9.51397, -8.21923,
-8.2106, -6.42983, -5.36826, -2.74515, -2.46561,
-2.31864, -2.23184, -1.5584, -1.54977, -0.988243,
-0.826667, -0.677152, -0.386428, 1.88659,
3.35719, 4.02123, 4.03768, 4.44442, 5.45841,
5.98133, 6.4066, 6.45027, 6.53159, 6.85724,
7.68041, 8.82803, 9.17761, 9.3086 Normal Scores
-2.04028, -1.60982, -1.36087, -1.17581,
-1.02411, -0.892918, -0.775547, -0.668002,
-0.567686, -0.472789, -0.381976, -0.294213,
-0.208664, -0.124617, -0.0414437, 0.0414437,
0.124617, 0.208664, 0.294213, 0.381976, 0.472789,
0.567686, 0.668002, 0.775547, 0.892918, 1.02411,
1.17581, 1.36087, 1.60982, 2.04028 n 30
Note Data was generated by a Uniform
Distribution on the interval -9,9.
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Example
Data 0.00881683, 0.295109, 2.71993, 0.0275762,
1.15885, 1.01363, 0.295519, 0.639201, 0.602931,
0.446441, 0.0801617, 0.580694, 0.367919,
0.477032, 0.197738, 0.16514, 1.43215, 0.305959,
0.269021, 0.359607 Sorted Data 0.00881683,
0.0275762, 0.0801617, 0.16514, 0.197738,
0.269021, 0.295109, 0.295519, 0.305959, 0.359607,
0.367919, 0.446441, 0.477032, 0.580694, 0.602931,
0.639201, 1.01363, 1.15885, 1.43215,
2.71993 Normal Scores -1.86824, -1.40341,
-1.12814, -0.919136, -0.744143, -0.589456,
-0.447768, -0.314572, -0.186756, -0.0619316,
0.0619316, 0.186756, 0.314572, 0.447768,
0.589456, 0.744143, 0.919136, 1.12814, 1.40341,
1.86824 n 20
Note Data was generated by a non-Normal
Distribution.
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The Normal Approximation to the Binomial
Probability Distribution
Section 7.5
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Example
According to the Commerce Department in 2004, 20
of U.S. households had some type of high-speed
internet connection (cable, DSL, satellite).
Suppose 80 U.S. households are selected at
random. What is the probability that exactly 15
households of the 80 will have a high-speed
internet connection?