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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
The Normal Distribution
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f(x)
x
Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
It can be shown:
[Area under the curve = 1]
[Mean of the distribution = ]
[Variance of the distribution = 2]
Parameters of the Normal Distribution
f x( )dx =1x!
( ) =x
dxxx f
( ) ( ) 22
dxx fxx
=
Holly Ott 8 Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
The net weight of nails in 20-lb boxes is normally distributed
with a mean () of 20 lbs and a variance (2) of 9 lbs2. Let the
random variable X represent the weights, then X ~ N(20, 9). We may
want to find the proportion of the boxes that have a net weight of
less than 15 lbs, the lower specification limit for net weights;
that is, we want P(X 15). This probability is given by the area
below 15 under the curve defined by the normal distribution
function with = 20lbs, 2 = 9lbs2.
Thus,
Normal Distribution: Example
Holly Ott 9 Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
= the proportion of the boxes that have a net weight of less
than 15 lbs.
Normal Distribution: Example
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But this is not easy to calculate!!
Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
A random variable that is normally distributed with with = 0 and
2 = 1 is called a Standard Normal Variable, denoted by Z. So, Z ~
N(0,1).
The Standard Normal Distribution
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(z)
z
Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
The pdf of Z is given by: And its CDF is given by: The table of
CDF for various values of z is called the Normal Table.
The Standard Normal Distribution
(z)= 12 e12 z2,< z
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
The Standard Normal Table
Holly Ott 13 Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
The Standard Normal Table
Holly Ott 14 Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
The Standard Normal Table
Holly Ott 15 Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
If Z ~ N(0,1) a) Find P(Z < 2.62) P(Z < 2.62) = 0.9956
The Standard Normal Table: Example
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
If Z ~ N(0,1) b) Find P(Z < -1.45 ) P(Z < -1.45 ) =
0.0735
The Standard Normal Table: Example
Holly Ott 17 Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
If Z ~ N(0,1) c) Find P(Z >1.45) P(Z > 1.45) = 1 P(Z <
1.45) = 1 0.9265 = 0.0735
The Standard Normal Table: Example
Holly Ott 18 Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
If Z ~ N(0,1) c) Find P(Z >1.45) P(Z > 1.45) = 1 P(Z <
1.45) = 1 0.9265 = 0.0735
The Standard Normal Table: Example
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b) Find P(Z < -1.45 ) P(Z < -1.45 ) = 0.0735
Quality Engineering & Management Module 3
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
If Z ~ N(0,1) d) Find P(-1.5 < Z < 2.5) P(-1.5 < Z <
2.5) = P(Z < 2.5) P(Z < -1.5) = 0.9938 0.0668 = 0.9270
The Standard Normal Table: Example
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TUM School of Management Production and Supply Chain Management
Prof Martin Grunow Technische Universitt Mnchen
Practice
Now let's do a practice problem to work on using the standard
normal tables.
Please complete the next "Practice" module in the course before
continuing with the lecture.
Holly Ott 21 Quality Engineering & Management Module 3
