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BA 201
Lecture 9
The Normal Distribution
© 2001 Prentice-Hall, Inc.
Chap 6-1
Topics

The Normal Distribution

The Standardized Normal Distribution
© 2001 Prentice-Hall, Inc.
Chap 6-2
Continuous Probability
Distributions

Continuous Random Variable



Values from interval of numbers
Absence of gaps
Continuous Probability Distribution


p.198
Distribution of continuous random variable
Most Important Continuous Probability
Distribution

The normal distribution
© 2001 Prentice-Hall, Inc.
Chap 6-3
p.199
The Normal Distribution





“Bell Shaped”
Symmetrical
Mean, Median and
Mode are Equal
Interquartile Range
Equals 1.33 s
Random Variable
has Infinite Range
© 2001 Prentice-Hall, Inc.
f(X)

X
Mean
Median
Mode
Chap 6-4
p.200
The Mathematical Model
(Called the Probability Density Function)
f X  

1
2s
2
X
e
1
2s X2
 X   X 2
f  X  : density of random variable X
  3.14159;
e  2.71828
 X : population mean
s X : population standard deviation
X : value of random variable    X   
© 2001 Prentice-Hall, Inc.
Chap 6-5
p.201
Many Normal Distributions
There are an Infinite Number of Normal Distributions
Varying the Parameters sX and X, We Obtain
Different Normal Distributions
© 2001 Prentice-Hall, Inc.
Chap 6-6
p.201
The Standardized Normal
Distribution

Given X
N   X ,s
2
X

, Z  X  X
has a
sX
standardized (normalized) normal distribution
where Z
f(Z)
N  0,1
sX
f(X)
sZ 1
X
© 2001 Prentice-Hall, Inc.
Z  0
X
Z
Chap 6-7
pp.203-212
Finding Probabilities
Probability is
the area under
the curve!
P c  X  d   ?
f(X)
a
© 2001 Prentice-Hall, Inc.
b
c
d
X
Chap 6-8
Example:
pp.203-212
P  2.9  X  7.1  .1664
Z
X  X
sX
2.9  5

 .21
10
Z
X  X
sX
7.1  5

 .21
10
Standardized
Normal Distribution
Normal Distribution
s X  10
.0832
sZ 1
.0832
2.9
X  5
© 2001 Prentice-Hall, Inc.
7.1
X
0.21
Shaded Area Exaggerated
Z  0
0.21
Z
Chap 6-9
Example:
pp.203-212
P  2.9  X  7.1  .1664(continued)
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
© 2001 Prentice-Hall, Inc.
0
Z = 0.21
Chap 6-10
Example:
pp.203-212
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
Shaded Area
Exaggerated
-0.2 .4207 .4168 .4129
-0.1 .4602 .4562 .4522
0.0 .5000 .4960 .4920
© 2001 Prentice-Hall, Inc.
0
Z = -0.21
Chap 6-11
p.247
Normal Distribution in PHStat


PHStat | Probability & Prob. Distributions |
Normal …
Example in Excel Spreadsheet
© 2001 Prentice-Hall, Inc.
Chap 6-12
Summary

Discussed the Normal Distribution

Described the Standard Normal Distribution
© 2001 Prentice-Hall, Inc.
Chap 6-13