Normal and t Distributions - University of Wisconsin–Madison
Normal and t Distributions
Bret Hanlon and Bret Larget
Department of Statistics
University of Wisconsin¡ªMadison
October 11¨C13, 2011
Normal
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Case Study
Case Study
Body temperature varies within individuals over time (it can be higher when
one is ill with a fever, or during or after physical exertion). However, if we
measure the body temperature of a single healthy person when at rest, these
measurements vary little from day to day, and we can associate with each
person an individual resting body temperture. There is, however, variation
among individuals of resting body temperture. A sample of n = 130 individuals had an average resting body temperature of 98.25 degrees Fahrenheit
and a standard deviation of 0.73 degrees Fahrenheit. The next slide shows
an estimated density plot from this sample.
Normal
Case Study
Body Temperature
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Density Plot
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Density
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Resting Body Temperature (F)
Normal
Case Study
Body Temperature
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Normal Distributions
The estimated density has these features:
I
I
it is bell-shaped;
it is nearly symmetric.
Many (but not all) biological variables have similar shapes.
One reason is a generalized the central limit theorem: random
variables that are formed by adding many random effects will be
approximately normally distributed.
Important for inference, even when underlying distributions are not
normal, the sampling distribution of the sample mean is
approximately normal.
Normal
Case Study
Body Temperature
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Example: Population
A population that is skewed.
Population
Density
0.006
0.004
0.002
0.000
0
200
400
600
x
Normal
Case Study
Body Temperature
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