Normal Distributions

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Normal Distributions
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Probability density function - the curved line
The height of the curve --gt density for a
particular X
Density relative concentration of observations
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The Normal Distribution
Symmetrical Bell Shape
Y
X
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The height of the curve at Xi
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sd 1
Mean
0
-3
-2
-1
0
1
2
3
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The Standardized Normal Curve --gt ? 0 and ? 1
Z
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?
50
50
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Lets say you have a population with a mean of
70kg mass and a standard deviation of 10 kg.
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?
50
50
70 kg
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?
?
?
70 kg
80 kg
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X
Z
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Standard Normal Deviate
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What does Z 1 mean?
Need to go to a table to get percent.
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70 kg
80 kg
X
0
1
Z
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Statistical Table 3 in Samuels and Witmer (sort
of)
Z

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Z0
Z1
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What can we say about this? Given a population
with a mean of 70 kg and a standard deviation of
10 kg, the probability of finding an individual
that is gt 80 kg in a random sample is 0.1587 (or
15.87).
We can also say.. Given a population with a
mean of 70 kg and a standard deviation of 10 kg,
the probability of finding an individual that is
lt 80 kg in a random sample is 1 - 0.1587 (or
84.13).
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?
84.13
15.87
70 kg
80 kg
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?
68
95
99.7
-1
1
0
2
3
-2
-3
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The CENTRAL LIMIT THEROEM So far, weve been
talking about populations. If we collect a BUNCH
of SAMPLES from a population having a normal
distribution ? the distribution of the MEANS of
those samples will also have a normal
distribution
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?25
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Frequency of means for forty samples of n 15
taken from a population comprised of N 5000
individuals having a mean of 25.
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Also, as the size of the samples increases, the
variance of the distributions will decrease.
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Variance of the Mean
If I collected all possible samples of size n and
calculated their means, the variance of the means
would equal the population variance divided by n.
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Standard Deviation of the Mean
This value is most commonly referred to as the
Standard Error of the Mean
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?
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So what?
Can answer What is the probability of
collecting a random sample of 10 individuals
that has a mean of greater than 80 kg in our
population that has a mean of 70 kg and a
standard deviation of 10 kg?
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?
84.13
15.87
70 kg
80 kg
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?
99.9
0.1
70 kg
80 kg