The Central Limit Theorem

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The Central Limit Theorem

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Sampling notation
From previous work we have the a population and a
sample and the following notation has been used.
Sample sample mean and sample variance

Population population mean
population variance
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Sampling
From previous work we know that if we take a
sample from a normal distribution the following
is true
and
Standard error

If we take a random sample of size n from any
distribution with a mean of and a variance of
then, for large n, the distribution of the
sum of the random variables is approximately
normal with a n and a variance of n .
This is called the central limit theorem.
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Central limit with a Binomial distribution
A game is played as follows an unbiased die is
rolled 18 times and the number of sixes is
recorded. A sample of 50 of these games is
played. Find the probability that the sample mean
is between 2.7 and 3.2.
Take the population and find the mean and
variance
np 3
and
npq 2.5
Use this to find the sample mean and standard
error
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s.e
These values can now be used with a normal
distribution curve.
Using a GDC (2.7,3.2,0.223,3)0.726 or by using
the z-figures from a table.
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Central limit with a Poisson distribution
In an insurance firm they have recorded an
average of 4 calls per hour. These calls are
independent and random. The company takes records
of 8 hours per day for 5 days, and records the
mean each hour. Find the probability that the
sample mean will exceed 4.5 calls.
Take the population and find the mean and
variance
Use this to find the sample mean and standard
error
These values can now be used with a normal
distribution curve.
Using a GDC (4.5,100,0.316,4)0.0568 or by using
the z-figures from a table.
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Central limit with any probability distribution
The central limit theorem works with probability
distribution (poisson, binomial, geometric,
hypergeometric, continuous models, etc). The
only stipulation is that n must be sufficiently
large.