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IE 440
PROCESS IMPROVEMENT
THROUGH PLANNED EXPERIMENTATION
The Normal Distribution and
Other Continuous Distributions
Dr. Xueping Li
University of Tennessee
© 2003 Prentice-Hall, Inc.
Chap 6-1
Chapter Topics

The Normal Distribution

The Standardized Normal Distribution

Evaluating the Normality Assumption

The Uniform Distribution

The Exponential Distribution
© 2003 Prentice-Hall, Inc.
Chap 6-2
Continuous Probability
Distributions

Continuous Random Variable



Continuous Probability Distribution


Values from interval of numbers
Absence of gaps
Distribution of continuous random variable
Most Important Continuous Probability
Distribution

The normal distribution
© 2003 Prentice-Hall, Inc.
Chap 6-3
The Normal Distribution





“Bell Shaped”
Symmetrical
Mean, Median and
Mode are Equal
Interquartile Range
Equals 1.33 s
Random Variable
Has Infinite Range
© 2003 Prentice-Hall, Inc.
f(X)

X
Mean
Median
Mode
Chap 6-4
The Mathematical Model
2
1
 (1/ 2)  X    / s 
f X  
e
2s
f  X  : density of random variable X
  3.14159;
e  2.71828
 : population mean
s : population standard deviation
X : value of random variable    X   
© 2003 Prentice-Hall, Inc.
Chap 6-5
Many Normal Distributions
There are an Infinite Number of Normal Distributions
Varying the Parameters s and , We Obtain
Different Normal Distributions
© 2003 Prentice-Hall, Inc.
Chap 6-6
The Standardized Normal
Distribution
When X is normally distributed with a mean  and a
standard deviation
s, Z 
X   follows a
s
standardized (normalized) normal distribution with a
mean 0 and a standard deviation 1.
f(Z)
s
f(X)
sZ 1

© 2003 Prentice-Hall, Inc.
Z  0
X
Z
Chap 6-7
Finding Probabilities
Probability is
the area under
the curve!
P c  X  d   ?
f(X)
c
© 2003 Prentice-Hall, Inc.
d
X
Chap 6-8
Which Table to Use?
Infinitely Many Normal Distributions
Means Infinitely Many Tables to Look Up!
© 2003 Prentice-Hall, Inc.
Chap 6-9
Solution: The Cumulative
Standardized Normal Distribution
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
sZ 1
.02
.5478
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
Probabilities
0.3 .6179 .6217 .6255
© 2003 Prentice-Hall, Inc.
0
Z = 0.12
Only One Table is Needed
Chap 6-10
Standardizing Example
Z
X 
s
Standardized
Normal Distribution
Normal Distribution
s  10
sZ 1
6.2
© 2003 Prentice-Hall, Inc.
6.2  5

 0.12
10
 5
X
0.12
Z  0
Z
Chap 6-11
Example:
P  2.9  X  7.1  .1664
Z
X 
s
2.9  5

 .21
10
Z
X 
s
7.1  5

 .21
10
Standardized
Normal Distribution
Normal Distribution
s  10
.0832
sZ 1
.0832
2.9 7.1
 5
© 2003 Prentice-Hall, Inc.
X
0.21 0.21
Z  0
Z
Chap 6-12
Example:
P  2.9  X  7.1  .1664(continued)
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
.02
sZ 1
.5832
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
© 2003 Prentice-Hall, Inc.
0
Z = 0.21
Chap 6-13
Example:
P  2.9  X  7.1  .1664(continued)
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
.02
Z  0
sZ 1
.4168
-0.3 .3821 .3783 .3745
-0.2 .4207 .4168 .4129
-0.1 .4602 .4562 .4522
0.0 .5000 .4960 .4920
© 2003 Prentice-Hall, Inc.
0
Z = -0.21
Chap 6-14
Normal Distribution in PHStat


PHStat | Probability & Prob. Distributions |
Normal …
Example in Excel Spreadsheet
© 2003 Prentice-Hall, Inc.
Chap 6-15
Example:
P  X  8  .3821
Z
X 
s
85

 .30
10
Standardized
Normal Distribution
Normal Distribution
s  10
sZ 1
.3821
 5
© 2003 Prentice-Hall, Inc.
8
X
0.30
Z  0
Z
Chap 6-16
Example:
P  X  8  .3821
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
.02
(continued)
sZ 1
.6179
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
© 2003 Prentice-Hall, Inc.
0
Z = 0.30
Chap 6-17
Finding Z Values for Known
Probabilities
What is Z Given
Probability = 0.6217 ?
Z  0
sZ 1
Cumulative Standardized
Normal Distribution Table
(Portion)
Z
.00
.01
0.2
0.0 .5000 .5040 .5080
.6217
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0
Z  .31
© 2003 Prentice-Hall, Inc.
0.3 .6179 .6217 .6255
Chap 6-18
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
© 2003 Prentice-Hall, Inc.
Chap 6-19
More Examples of Normal
Distribution Using PHStat
A set of final exam grades was found to be normally
distributed with a mean of 73 and a standard deviation of 8.
What is the probability of getting a grade no higher than 91
on this exam?
X
N  73,8
2

Mean
Standard Deviation
P  X  91  ?
73
8
Probability for X <=
X Value
91
Z Value
2.25
P(X<=91)
0.9877756
© 2003 Prentice-Hall, Inc.
s 8
X
  73 91
0
2.25
Z
Chap 6-20
More Examples of Normal
Distribution Using PHStat
(continued)
What percentage of students scored between
65 and 89?
X
N  73,82 
P  65  X  89  ?
Probability for a Range
From X Value
65
To X Value
89
Z Value for 65
-1
Z Value for 89
2
P(X<=65)
0.1587
P(X<=89)
0.9772
P(65<=X<=89)
0.8186
X
65
  73 89
-1 0
© 2003 Prentice-Hall, Inc.
Z
2
Chap 6-21
More Examples of Normal
Distribution Using PHStat
(continued)
Only 5% of the students taking the test
scored higher than what grade?
X
N  73,8
2

P  ?  X   .05
Find X and Z Given Cum. Pctage.
Cumulative Percentage
95.00%
Z Value
1.644853
X Value
86.15882
X
  73 ? =86.16
0
© 2003 Prentice-Hall, Inc.
Z
1.645
Chap 6-22
More Examples of Normal
Distribution Using PHStat
(continued)
The middle 50% of the students scored
between what two scores?
X
N  73,82 
P  a  X  b  .50
Find X and Z Given Cum. Pctage.
Cumulative Percentage
25.00%
Z Value
-0.67449
X Value
67.60408
Find X and Z Given Cum. Pctage.
Cumulative Percentage
75.00%
Z Value
0.67449
X Value
78.39592
© 2003 Prentice-Hall, Inc.
.25
.25
X
67.6   73 78.4
-0.67 0
Z
0.67
Chap 6-23
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
© 2003 Prentice-Hall, Inc.
Chap 6-24
Assessing Normality

Construct Charts



(continued)
For small- or moderate-sized data sets, do the
stem-and-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?
© 2003 Prentice-Hall, Inc.
Chap 6-25
Assessing Normality

Observe the Distribution of the Data Set




(continued)
Do approximately
between mean 
Do approximately
between mean 
Do approximately
between mean 
2/3 of the observations lie
1 standard deviation?
4/5 of the observations lie
1.28 standard deviations?
19/20 of the observations lie
2 standard deviations?
Evaluate Normal Probability Plot

Do the points lie on or close to a straight line with
positive slope?
© 2003 Prentice-Hall, Inc.
Chap 6-26
Assessing Normality

(continued)
Normal Probability Plot




Arrange Data into Ordered Array
Find Corresponding Standardized Normal Quantile
Values
Plot the Pairs of Points with Observed Data Values
on the Vertical Axis and the Standardized Normal
Quantile Values on the Horizontal Axis
Evaluate the Plot for Evidence of Linearity
© 2003 Prentice-Hall, Inc.
Chap 6-27
Assessing Normality
(continued)
Normal Probability Plot for Normal
Distribution
90
X 60
Z
30
-2 -1 0 1 2
© 2003 Prentice-Hall, Inc.
Look for Straight Line!
Chap 6-28
Normal Probability Plot
Left-Skewed
Right-Skewed
90
90
X 60
X 60
Z
30
-2 -1 0 1 2
-2 -1 0 1 2
Rectangular
U-Shaped
90
90
X 60
X 60
Z
30
-2 -1 0 1 2
© 2003 Prentice-Hall, Inc.
Z
30
Z
30
-2 -1 0 1 2
Chap 6-29
Obtaining Normal Probability
Plot in PHStat


PHStat | Probability & Prob. Distributions |
Normal Probability Plot
Enter the range of the cells that contain the
data in the Variable Cell Range window
© 2003 Prentice-Hall, Inc.
Chap 6-30
The Uniform Distribution

Properties:


The probability of occurrence of a value is equally
likely to occur anywhere in the range between the
smallest value a and the largest value b
Also called the rectangular distribution


© 2003 Prentice-Hall, Inc.

a  b
2
2
b  a 
2
s 
12
Chap 6-31
The Uniform Distribution
(continued)

The Probability Density Function
1
f X  
if a  X  b
b  a 


Application: Selection of random numbers
E.g., A wooden wheel is spun on a horizontal
surface and allowed to come to rest. What is
the probability that a mark on the wheel will
point to somewhere between the North and
the East?
90
P  0  X  90  
 0.25
360
© 2003 Prentice-Hall, Inc.
Chap 6-32
Exponential Distributions
P  arrival time  X   1  e
 X
X : any value of continuous random variable
 : the population average number of
arrivals per unit of time
1/: average time between arrivals
e  2.71828
E.g., Drivers arriving at a toll bridge;
customers arriving at an ATM machine
© 2003 Prentice-Hall, Inc.
Chap 6-33
Exponential Distributions
(continued)

Describes Time or Distance between Events


f(X)
Density Function


Used for queues
f  x 
Parameters

 
© 2003 Prentice-Hall, Inc.
1

e

x

 = 0.5
 = 2.0
X
s 
Chap 6-34
Example
E.g., Customers arrive at the checkout line
of a supermarket at the rate of 30 per hour.
What is the probability that the arrival time
between consecutive customers will be
greater than 5 minutes?
  30
X  5 / 60 hours
P  arrival time >X   1  P  arrival time  X 

 1 1 e
30 5/ 60 

 .0821
© 2003 Prentice-Hall, Inc.
Chap 6-35
Exponential Distribution
in PHStat


PHStat | Probability & Prob. Distributions |
Exponential
Example in Excel Spreadsheet
© 2003 Prentice-Hall, Inc.
Chap 6-36
Chapter Summary

Discussed the Normal Distribution

Described the Standard Normal Distribution

Evaluated the Normality Assumption

Defined the Uniform Distribution

Described the Exponential Distribution
© 2003 Prentice-Hall, Inc.
Chap 6-37