Download Statistics Sampling Distribution Note

9/30/2015
Statistics
• A statistic is any quantity whose value
can be calculated from sample data.
CH5: Statistics and their
distributions
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• A statistic can be thought of as a random
variable.
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Sampling Distribution
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Note
• Any statistic, being a random variable, has
a probability distribution.
• The probability distribution of a statistic is
sometimes referred to as its sampling
distribution.
• In this group of notes we will look at
examples where we know the population
and it’s parameters.
• This is to give us insight into how to
proceed when we have large populations
with unknown parameters (which is the
more typical scenario).
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The “Meta-Experiment”
Sample Statistics
• The “Meta-Experiment” consists of indefinitely
many repetitions of the same experiment.
• If the experiment is taking a sample of 100 items
from a population, the meta-experiment is to
repeatedly take samples of 100 items from the
population.
• This is a theoretical construct to help us
understand the probabilities involved in our
experiment.
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Distribution of the Sample Mean
Meta-Experiment
Experiment
Population
Sample
Population
Statistic
Sample
of n
Sample
Statistic
Sample of n
Sample of n
Sample
Statistic
Sample of n
Sample
. Statistic
.
Etc.
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Example: Random Rectangles
100 Rectangles with µ=7.42 and σ=5.26.
Let X1, X2,…,Xn be a random sample from
a distribution with mean value µ and
standard deviation σ. Then
frequency
1. E ( X )   X  
2. V ( X )   X2   2 / n
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SD ( X )   X   / n
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Histogram of Areas
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Areas
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Based on 68 random samples of size 5:
Mean of the sample means=7.33
SD of the sample means=1.88.
So, the distribution of the sample mean
based on samples of size 5, should have
1. E ( X )  7.42
Histogram of Sample Means
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0
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frequency
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2. SD ( X )  5.26 / 5  2.35
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Means
Normal Distributions
Example: Women’s Heights
• Let X1, X2,…,Xn be a random sample from
a normal distribution with mean value µ
and standard deviation σ.
• Then for any n, X is normally distributed
(with mean µ and standard deviation  / n ).
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• It is known that women’s heights are normally
distributed with population mean 64.5 inches
and population standard deviation 2.5 inches.
• We will look at the distribution of sample
means for various sample sizes.
• Since the population follows a normal
distribution, the sampling distribution of X is
also normal regardless of sample size.
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For n=9, the sample means will be normally distributed with
mean=64.5 and standard deviation= 2.5 / 9  0.83.
For n=25, the sample means will be normally distributed with
mean=64.5 and standard deviation= 2.5 / 25  0.5.
Distribution of Sample Means (n=25)
0.2
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Height of Curve
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Height of Curve
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Distribution of Sample Means (n=9)
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Sample
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For n=100, the sample means will be normally distributed with
mean=64.5 and standard deviation = 2.5 / 100  0.25.
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Sample Mean
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The Central Limit Theorem
Distribution of Sample Means (n=100)
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E ( X )   and V ( X )   2 / n.
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Height of Curve
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• Let X1, X2,…,Xn be a random sample from
a distribution with mean value µ and
standard deviation σ.
• Then if n is sufficiently large, X has
approximately a normal distribution with
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Sample Mean
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Rule of Thumb
Sampling Distribution Simulation
• If n>30, the Central Limit Theorem can be
used.
http://onlinestatbook.com/stat_sim/index.html
• For highly skewed data or data with
extreme outliers, it may take sample of
40+ before the CLT starts “working”.
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The sampling distribution of X depends on the
sample size (n) in two ways:
1. The standard deviation is  X   / n
which is inversely proportional to n
2. If the population distribution is not normal, then
the shape of the sampling distribution of
X depends on n, being more nearly normal
for larger n.
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Why is the Central Limit Theorem
Important?
Dependence on Sample Size
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• Every different type of population needs a
different set of procedures (i.e. probability
tables) to answer questions about probability.
• The CLT shows us that we can use the same
procedure (normal probability procedures) for
questions about probability and the sample
mean, regardless of the shape of the original
population.
• The only requirement is a “large” sample.
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Proportions as Means
Distribution of Sample Proportion
• Recall that a binomial RV X is the number
of successes in an experiment consisting
of n independent success/failure trials.
• Let
1 if the ith trial results in a success
• The CLT implies that if n is sufficiently
large, then p̂ has approximately a normal
distribution with
p (1  p )
n
where p is the true population proportion.
E ( pˆ )  p and V ( pˆ ) 
Xi  
0 if the ith trial results in a failure
• Then the sample proportion can be
expressed as
pˆ 
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
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X
i
n
• In order for the approximation to hold we need
np≥10 and n(1-p)≥10.
X
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Example: Suppose we took samples of size 20
from a population where p=0.5.
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Probability
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The sampling distribution of p̂ is approximately
Normal with mean = 0.5 and
variance = p(1  p ) / n  0.5  0.5 / 20  0.0125
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p-hat
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