© 2003 prentice-hall, inc.chap 6-1 the normal distribution and other continuous distributions ie...
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© 2003 Prentice-Hall, Inc. Chap 6-1
The Normal Distribution and Other Continuous
Distributions
IE 440
PROCESS IMPROVEMENT THROUGH PLANNED EXPERIMENTATION
Dr. Xueping LiUniversity of Tennessee
© 2003 Prentice-Hall, Inc. Chap 6-2
Chapter Topics
The Normal Distribution The Standardized Normal Distribution Evaluating the Normality Assumption The Uniform Distribution The Exponential Distribution
© 2003 Prentice-Hall, Inc. Chap 6-3
Continuous Probability Distributions
Continuous Random Variable Values from interval of numbers Absence of gaps
Continuous Probability Distribution Distribution of continuous random variable
Most Important Continuous Probability Distribution The normal distribution
© 2003 Prentice-Hall, Inc. Chap 6-4
The Normal Distribution
“Bell Shaped” Symmetrical Mean, Median and
Mode are Equal Interquartile Range
Equals 1.33 Random Variable
Has Infinite Range
Mean Median Mode
X
f(X)
© 2003 Prentice-Hall, Inc. Chap 6-5
The Mathematical Model
2(1/ 2) /1
2
: density of random variable
3.14159; 2.71828
: population mean
: population standard deviation
: value of random variable
Xf X e
f X X
e
X X
© 2003 Prentice-Hall, Inc. Chap 6-6
Many Normal Distributions
Varying the Parameters and , We Obtain Different Normal Distributions
There are an Infinite Number of Normal Distributions
© 2003 Prentice-Hall, Inc. Chap 6-7
The Standardized Normal Distribution
When X is normally distributed with a mean
and a standard deviation , follows
a standardized (normalized) normal distribution
with a mean 0 and a standard deviation 1.
XZ
X
f(X)
Z
0Z
1Z
f(Z)
© 2003 Prentice-Hall, Inc. Chap 6-8
Finding Probabilities
Probability is the area under the curve!
c dX
f(X)
?P c X d
© 2003 Prentice-Hall, Inc. Chap 6-9
Which Table to Use?
Infinitely Many Normal Distributions Means Infinitely Many Tables to Look
Up!
© 2003 Prentice-Hall, Inc. Chap 6-10
Solution: The Cumulative Standardized Normal
Distribution
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.5478.02
0.1 .5478
Cumulative Standardized Normal Distribution Table (Portion)
Probabilities
Only One Table is Needed
0 1Z Z
Z = 0.12
0
© 2003 Prentice-Hall, Inc. Chap 6-11
Standardizing Example
6.2 50.12
10
XZ
Normal Distribution
Standardized Normal
Distribution10 1Z
5 6.2 X Z
0Z 0.12
© 2003 Prentice-Hall, Inc. Chap 6-12
Example:
Normal Distribution
Standardized Normal
Distribution10 1Z
5 7.1 X Z0Z
0.21
2.9 5 7.1 5.21 .21
10 10
X XZ Z
2.9 0.21
.0832
2.9 7.1 .1664P X
.0832
© 2003 Prentice-Hall, Inc. Chap 6-13
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.5832.02
0.1 .5478
Cumulative Standardized Normal Distribution Table (Portion)
0 1Z Z
Z = 0.21
Example: 2.9 7.1 .1664P X
(continued)
0
© 2003 Prentice-Hall, Inc. Chap 6-14
Z .00 .01
-0.3 .3821 .3783 .3745
.4207 .4168
-0.1.4602 .4562 .4522
0.0 .5000 .4960 .4920
.4168.02
-0.2 .4129
Cumulative Standardized Normal Distribution Table (Portion)
0 1Z Z
Z = -0.21
Example: 2.9 7.1 .1664P X
(continued)
0
© 2003 Prentice-Hall, Inc. Chap 6-15
Normal Distribution in PHStat
PHStat | Probability & Prob. Distributions | Normal …
Example in Excel Spreadsheet
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 6-16
Example: 8 .3821P X
Normal Distribution
Standardized Normal
Distribution10 1Z
5 8 X Z0Z
0.30
8 5.30
10
XZ
.3821
© 2003 Prentice-Hall, Inc. Chap 6-17
Example: 8 .3821P X
(continued)
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.6179.02
0.1 .5478
Cumulative Standardized Normal Distribution Table (Portion)
0 1Z Z
Z = 0.30
0
© 2003 Prentice-Hall, Inc. Chap 6-18
.6217
Finding Z Values for Known Probabilities
Z .00 0.2
0.0 .5000 .5040 .5080
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
.6179 .6255
.01
0.3
Cumulative Standardized Normal Distribution Table
(Portion)
What is Z Given Probability = 0.6217 ?
.6217
0 1Z Z
.31Z 0
© 2003 Prentice-Hall, Inc. Chap 6-19
Recovering X Values for Known Probabilities
5 .30 10 8X Z
Normal Distribution
Standardized Normal
Distribution10 1Z
5 ? X Z0Z 0.30
.3821.6179
© 2003 Prentice-Hall, Inc. Chap 6-20
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?
273,8X N 91 ?P X Mean 73Standard Deviation 8
X Value 91Z Value 2.25P(X<=91) 0.9877756
Probability for X <=
2.250
X
Z91
8
73
© 2003 Prentice-Hall, Inc. Chap 6-21
What percentage of students scored between 65 and 89?
From X Value 65To X Value 89Z Value for 65 -1Z Value for 89 2P(X<=65) 0.1587P(X<=89) 0.9772P(65<=X<=89) 0.8186
Probability for a Range
273,8X N 65 89 ?P X
20
X
Z8965
-1
73
More Examples of Normal Distribution Using PHStat
(continued)
© 2003 Prentice-Hall, Inc. Chap 6-22
73
Only 5% of the students taking the test scored higher than what grade?
273,8X N ? .05P X
Cumulative Percentage 95.00%Z Value 1.644853X Value 86.15882
Find X and Z Given Cum. Pctage.
1.6450
X
Z? =86.16
(continued)
More Examples of Normal Distribution Using PHStat
© 2003 Prentice-Hall, Inc. Chap 6-23
The middle 50% of the students scored between what two scores?
273,8X N
Cumulative Percentage 75.00%Z Value 0.67449X Value 78.39592
Find X and Z Given Cum. Pctage.
Cumulative Percentage 25.00%Z Value -0.67449X Value 67.60408
Find X and Z Given Cum. Pctage.
0.670
X
Z78.467.6
-0.67
.50P a X b
.25.25
73
More Examples of Normal Distribution Using PHStat
(continued)
© 2003 Prentice-Hall, Inc. Chap 6-24
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 6-25
Assessing Normality Construct Charts
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
? Is the range approximately 6 ?
(continued)
© 2003 Prentice-Hall, Inc. Chap 6-26
Assessing Normality
Observe the Distribution of the Data Set 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? Evaluate Normal Probability Plot
Do the points lie on or close to a straight line with positive slope?
(continued)
© 2003 Prentice-Hall, Inc. Chap 6-27
Assessing Normality
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
(continued)
© 2003 Prentice-Hall, Inc. Chap 6-28
Assessing Normality
Normal Probability Plot for Normal Distribution
Look for Straight Line!
30
60
90
-2 -1 0 1 2
Z
X
(continued)
© 2003 Prentice-Hall, Inc. Chap 6-29
Normal Probability Plot
Left-Skewed Right-Skewed
Rectangular U-Shaped
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
© 2003 Prentice-Hall, Inc. Chap 6-30
Obtaining Normal ProbabilityPlot 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 6-31
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
2
a b
2
2
12
b a
© 2003 Prentice-Hall, Inc. Chap 6-32
The Uniform Distribution The Probability Density Function
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?
(continued)
1
if f X a X bb a
900 90 0.25
360P X
© 2003 Prentice-Hall, Inc. Chap 6-33
Exponential Distributions
arrival time 1
: any value of continuous random variable
: the population average number of
arrivals per unit of time
1/ : average time between arrivals
2.71828
XP X e
X
e
E.g., Drivers arriving at a toll bridge; customers arriving at an ATM machine
© 2003 Prentice-Hall, Inc. Chap 6-34
Exponential Distributions
Describes Time or Distance between Events Used for queues
Density Function
Parameters
(continued)
f(X)
X
= 0.5
= 2.0
1 x
f x e
© 2003 Prentice-Hall, Inc. Chap 6-35
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 5/ 60
30 5 / 60 hours
arrival time > 1 arrival time
1 1
.0821
X
P X P X
e
© 2003 Prentice-Hall, Inc. Chap 6-36
Exponential Distributionin PHStat
PHStat | Probability & Prob. Distributions | Exponential
Example in Excel Spreadsheet
Microsoft Excel Worksheet
© 2003 Prentice-Hall, Inc. Chap 6-37
Chapter Summary
Discussed the Normal Distribution
Described the Standard Normal
Distribution
Evaluated the Normality Assumption
Defined the Uniform Distribution
Described the Exponential Distribution
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