Transcript PPT

Chapter 3 ~ Normal Probability Distributions
P(a  x  b)
a

b
x
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Chapter Goals
• Learn about the normal, bell-shaped, or
Gaussian distribution
• How probabilities are found
• How probabilities are represented
• How normal distributions are used in the real
world
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Normal Probability Distributions
• The normal probability distribution is the most
important distribution in all of statistics
• Many continuous random variables have normal
or approximately normal distributions
• Need to learn how to describe a normal
probability distribution
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Normal Probability Distribution
1. A continuous random variable
2. Description involves two functions:
a. A function to determine the ordinates of the graph
picturing the distribution
b. A function to determine probabilities
3. Normal probability distribution function:
1 ( x-)
e 2 s
2
1
s 2p
This is the function for the normal (bell-shaped) curve
f ( x) =
4. The probability that x lies in some interval is the area
under the curve
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The Normal Probability Distribution
s
 - 3s  - 2s  - s

 s
  2s   3s
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Probabilities for a Normal Distribution
• Illustration
b
P(a  x  b) =  f ( x )dx
a
a
b
x
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Notes

The definite integral is a calculus topic

We will use a table to find probabilities for normal
distributions

We will learn how to compute probabilities for one special
normal distribution: the standard normal distribution

Transform all other normal probability questions to this
special distribution

Recall the empirical rule: the percentages that lie within
certain intervals about the mean come from the normal
probability distribution

We need to refine the empirical rule to be able to find the
percentage that lies between any two numbers
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Percentage, Proportion & Probability
• Basically the same concepts
• Percentage (30%) is usually used when talking
about a proportion (3/10) of a population
• Probability is usually used when talking about
the chance that the next individual item will
possess a certain property
• Area is the graphic representation of all three
when we draw a picture to illustrate the situation
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The Standard Normal Distribution
• There are infinitely many normal probability
distributions
• They are all related to the standard normal
distribution
• The standard normal distribution is the
normal distribution of the standard variable z
(the z-score)
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Standard Normal Distribution
Properties:
• The total area under the normal curve is equal to 1
• The distribution is mounded and symmetric; it extends indefinitely in
both directions, approaching but never touching the horizontal axis
• The distribution has a mean of 0 and a standard deviation of 1
• The mean divides the area in half, 0.50 on each side
• Nearly all the area is between z = -3.00 and z = 3.00
Notes:

Appendix, Table A lists the probabilities associated with the intervals
from the mean (0) to a specific value of z

Probabilities of other intervals are found using the table
entries, addition, subtraction, and the properties above
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Which Table to Use?
An infinite number of normal distributions
means an infinite number of tables to look up!
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Solution: The Cumulative
Standardized Normal Distribution
Cumulative Standardized Normal
Distribution Table (Portion)
Z = 0
Z
.00
.01
sZ =1
.02
.5478
0.0 .5000 .5040 .5080
Shaded Area
Exaggerated
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
Probabilities
0.3 .6179 .6217 .6255
0
Z = 0.12
Only One Table is Needed
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Standardizing Example
Z=
X -
s
6.2 - 5
=
= 0.12
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Standardized
Normal Distribution
Normal Distribution
s = 10
 =5
sZ =1
6.2
X
Shaded Area Exaggerated
Z = 0
0.12
Z
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Example:
P  2.9  X  7.1 = .1664
Z=
X -
s
2.9 - 5
=
= -.21
10
Z=
X -
s
7.1 - 5
=
= .21
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Standardized
Normal Distribution
Normal Distribution
s = 10
.0832
sZ =1
.0832
2.9
7.1
X
-0.21
Shaded Area Exaggerated
Z = 0
0.21
Z
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Example:
P  2.9  X  7.1 = .1664
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z = 0
sZ =1
.02
.5832
0.0 .5000 .5040 .5080
Shaded Area
Exaggerated
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
(continued)
0
Z = 0.21
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Example:
P  2.9  X  7.1 = .1664
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
.02
Z = 0
sZ =1
.4168
-03 .3821 .3783 .3745
Shaded Area
Exaggerated
-02 .4207 .4168 .4129
-0.1 .4602 .4562 .4522
0.0 .5000 .4960 .4920
(continued)
0
Z = -0.21
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Notes

The symmetry of the normal distribution is a key factor
in determining probabilities associated with values
below (to the left of) the mean. For example: the area
between the mean and z = -1.37 is exactly the same as
the area between the mean and z = +1.37.

When finding normal distribution probabilities, a sketch
is always helpful. See course worksheet.
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Example:
P  X  8 = .3821
Z=
X -
s
8-5
=
= .30
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Standardized
Normal Distribution
Normal Distribution
s = 10
sZ =1
.3821
 =5
8
X
Shaded Area Exaggerated
Z = 0
0.30
Z
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Example:
P  X  8 = .3821
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z = 0
(continued)
sZ =1
.02
.6179
0.0 .5000 .5040 .5080
Shaded Area
Exaggerated
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
0
Z = 0.30
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Recovering X Values for Known
Probabilities
Standardized
Normal Distribution
Normal Distribution
s = 10
sZ =1
.6179
.3821
 =5
?
X
Z = 0
0.30
Z
X =   Zs = 5  .3010 = 8
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Assessing Normality
• Not all continuous random variables are normally
distributed
• It is important to evaluate how well the data set
seems to be adequately approximated by a normal
distribution
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Assessing Normality
(continued)
• Construct charts
– For small- or moderate-sized data sets, do stemand-leaf display and box-and-whisker plot look
symmetric?
– For large data sets, does the histogram or polygon
appear bell-shaped?
• Compute descriptive summary measures
– Do the mean, median and mode have similar
values?
– Is the interquartile range approximately 1.33 s?
– Is the range approximately 6 s?
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Assessing Normality
• Observe the distribution of the data set
(continued)
– Do approximately 2/3 of the observations lie
between mean  1 standard deviation?
– Do approximately 4/5 of the observations lie
between mean  1.28 standard deviations?
– Do approximately 19/20 of the observations lie
between mean  2 standard deviations?
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Assessing Normality
• With Minitab
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Applications of Normal Distributions
• Apply the techniques learned for the z distribution
to all normal distributions
• Start with a probability question in terms of
x-values
• Convert, or transform, the question into an
equivalent probability statement involving
z-values
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Standardization
• Suppose x is a normal random variable with mean  and
standard deviation s
x-
• The random variable z =
s
distribution

0
has a standard normal
c
c-
s
x
z
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Notes
• The normal table may be used to answer many kinds of questions
involving a normal distribution
• Often we need to find a cutoff point: a value of x such that there is
a certain probability in a specified interval defined by x
 Example: The waiting time x at a certain bank is approximately
normally distributed with a mean of 3.7 minutes and a
standard deviation of 1.4 minutes. The bank would
like to claim that 95% of all customers are waited on
by a teller within c minutes. Find the value of c that
makes this statement true.
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Solution
0.0500
0.5000 0.4500
3.7
0
P ( x  c) = 0.95
 x - 3.7  c - 3.7  =
 0.95
P
 14
.
1.4 
  c - 3.7  =
 0.95
P z

14
. 
c
1645
.
x
z
c - 3.7
= 1645
.
14
.
c = (1645
. )(14
. )  3.7 = 6.003
c  6 minutes
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Notation
• If x is a normal random variable with mean  and
standard deviation s, this is often denoted:
x ~ N(, s)

Example: Suppose x is a normal random variable
with  = 35 and s = 6. A convenient notation to
identify this random variable is: x ~ N(35, 6).
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Notation
• z-score used throughout statistics in a variety of
ways
• Need convenient notation to indicate the area
under the standard normal distribution
• z(a) is the algebraic name, for the z-score (point on
the z axis) such that there is a of the area
(probability) to the right or left of z(a)
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Notes
• The values of z that will be used regularly come from
one of the following situations:
1. The z-score such that there is a specified area in one
tail of the normal distribution
2. The z-scores that bound a specified middle
proportion of the normal distribution
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Example
 Example: Find the numerical value of z(0.99):
0.01
z(0.01)
0
z
• Because of the symmetrical nature of the normal distribution,
z(0.99) = -z(0.01)
• Using Table A: z(0.01) = -2.33
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Example
 Example: Find the z-scores that bound the middle 0.99 of the
normal distribution:
0.005
0.005
0.495
z(0.005)
0.495
0
z(0.995)
• Use Table A:
z(0.005) = -2.575 and z(0.995) = 2.575
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Chapter Summary
• Discussed the normal distribution
• Described the standard normal distribution
• Evaluated the normality assumption
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