Download Probability Theory and the Magic of the Normal Distribution


Normal Curves
 The family of normal curves
 The rule of 68-95-99.7

The Central Limit Theorem

Confidence Intervals
 Around a Mean
 Around a Proportion

Normal curves are a special family
of density curves, which are graphs
that answer the question: “what
proportion of my cases take on
values that fall within a certain
range?”
 Many things in nature, such as sizes of animals and
errors in astronomic calculations, happen to be
normally distributed.
.4
.3
.3
.3
.2
.1
.1
.1
.0
0.0
< 57.5
.0
60 - 62.5
57.5 - 60
Height in Inches
65 - 67.5
62.5 - 65
72.5 - 75
67.5 - 70
> 75

What do all normal
curves have in
common?

How can we tell
one normal curve
from another?
 Symmetric
 Mean tells you where
 Mean = Median
it is centered
 Standard deviation
tells you how thick or
narrow the curve will
be
 Bell-shaped, with
most of their density
in center and little in
the tails

The 68-95-99.7 Rule.
 68% of cases will take on a value that is plus
or minus one standard deviation of the mean
 95% of cases will take on a value that is plus
or minus two standard deviations
 99.7% of cases will take on a value that is plus
or minus three standard deviations

If we take repeated samples from a
population, the sample means will be
(approximately) normally distributed.
 The mean of the “sampling distribution”
will equal the true population mean.
 The “standard error” (the standard
deviation of the sampling distribution)
equals 
N

A “sampling distribution” of a statistic tells us what
values the statistic takes in repeated samples from
the same population and how often it takes them.

We use the statistical properties of a
distribution of many samples to see how
confident we are that a sample statistic is
close to the population parameter
 We can compute a confidence interval around
a sample mean or a proportion
 We can pick how confident we want to be
 Usually choose 95%, or two standard errors

The 95% confidence interval around a
sample mean is:
X  2
̂
N
whereˆ 
( X  X )
i
N 1
2

If my sample of 100 donors finds a mean
contribution level of $15,600 and I
compute a confidence interval that is:
$15,600 + or - $600

I can make the statement: I can say at the
95% confidence level that the mean
contribution for all donors is between
$15,000 and $16,200.

The 95% confidence interval around a
sample proportion is:
pˆ 
2
( pˆ )(1  pˆ )
N
And the 99.7% confidence interval would be:
pˆ 
3
( pˆ )(1  pˆ )
N

The margin of error is calculated by:
pˆ 
2
( pˆ )(1  pˆ )
N

In a poll of 505 likely voters, the Field
Poll found 55% support for a
constitutional convention.
0.55 
2
(0.55)(1  0.55)
505
0.55  0.044

The margin of error for this poll was plus
or minus 4.4 percentage points.

This means that if we took many samples
using the Field Poll’s methods, 95% of
the samples would yield a statistic within
plus or minus 4.4 percentage points of the
true population parameter.