16 March 2011
IE 256 STATISTICS FOR INDUSTRIAL ENGINEERS
PS # 3
In this PS, we have discussed the theoretical background of sampling distributions for the mean.
Then we considered the first question of PS # 2 to have a practical insight. Lastly, we have
concluded the PS with a simulation of these discussions in a statistical package namely R. You
are not responsible with the coding procedure in R, I use it just to show you some graphical
results.
The following figure shows the sampling process from a population. A population is the
collection of all possible outcomes of a random variable
ܺ
. Each individual in the population,
represented by
, follows a distribution
. The mean of the population is
and the
standard deviation is
ߪ
. If all outcomes in the population with their probabilities are not given or
the distribution of the population with parameters of the distribution is not known we cannot
calculate the values of
and
. Since generally the size of the population is very large, we
cannot figure out all outcomes in the population from practical point of view. Hence, we decide
to make measurements for a limited number of trials, say
݊
many outcomes, and constitute a
sample of the population.
ܺ
ܦ݅ݏݐ
ଵ
ߤ
ߤ
ߪ
1
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Each outcome in a sample, represented with
ܺ
,݅ൌ1
,…
,݊
, comes from
ܦ݅ݏݐ
ଵ
with same
parameter values, since they are members of the population. We know all values in a sample,
therefore we can calculate sample mean
ܺ
ത
and sample standard deviation
ܵ
. Let us repeat this
sampling process for
݉
times. This means we take
݉
different samples each of which has
݊
observations. Since we are taking these
݉
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