1 normal distribution
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HYPOTHESIS
TESTING- NORMALDISTRIBUTION
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is symmetrical about the mean.
is asymptotic. That is the curve gets closer and
closer to theX-axis but never actually touches it.Has its mean, , to determine its location and itsstandard deviation, , to determine its dispersion.
The Normal probability distribution
is bell-shaped and has a single peak at the
center of the distribution.
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5
0
.
4
0
.
3
0
.
2
0
.
1
.
0
x
f
(
x
r
a l
i
t
r b u i
o n :
m = 0 ,
s2 = 1
Mean, median, and
mode are equal
Theoretically, curve
extends to
infinity
a
Characteristics of a Normal Distribution
Normal curve issymmetrical
7-10
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Standard Normal (Z) Table
There are infinite number of combination of and exist, an infinite number of normal
distributions exist and an infinite number of
tables would be required.
However, by standardizing the data, we need
only one table- Z table
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The Standard NormalProbability Distribution
Xz
It is also called the
z distribution.
The standardnormal distributionis a normal distribution
with a mean of 0 and astandard deviation of 1.
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Uses of normal distribution
Finding probabilities corresponding to knownvalue of X or Z
Given X or Z value: find the probability of its
occurrence Finding values of X or Z corresponding to
known probabilities
Given the probability of occurrence: find the X orZ that would occur
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Finding probability corresponding toknown values of X or Z
Example: An engineering firms contracts receivedper year follows a normal distribution with = 50and = 5
Area under a normal bell shaped curve for this firmrelate to the entire history of the firm representing thepopulation
The probabilities or proportion of the area under the
curve must add up to 1
Q) Firm wants to know what is the probability that itwill receive between 50 - 55 contracts next year?
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Finding probability correspondingto known values of X or Z
Converting to standard normal value
From the Z table for Z=1.00 the associatedprobability = 0.3413
Az-value of 1 indicates that the value of 55 isone standard deviation above the mean of 50.
00.1
5
5055
Z
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Finding Probabilities Corresponding to KnownValues
-3 -2 -1 +1 +2 +3
35
-3
40
-2
45
-1
50
0
55
+1
60
+2
65
+3
Area is 0.3413
Z Scale
Z Scale
(=50, =5)
Area between and + 1= 0.3431
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Finding probability correspondingto known values of X or Z
If the firm wants to find out what is theprobability that it will get between 45 to 50contracts next year?
Z = -1.00
Area between and - 1 = -0.3431
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Finding probability correspondingto known values of X or Z
What is the probability for the firm to getcontracts between 45 and 55 next year?
Area between 1 = (2 x 0.3431) = 0.6826
Similarly area between 2 = 0.9544
Area between 3 = 0.9973
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Areas Under the Normal
Curve
Practically all is within three standard
deviations of the mean.
+ 3
About 68 percent of
the area under the
normal curve is withinone standard deviation
of the mean.
+ 1
About 95 percent is within two standard
deviations of the mean.
+ 2