Download sampling distribution

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Sampling Distributions and Inference for
Proportions(C18-C22 BVD)
C18: Sampling Distributions
* The sampling distribution of a statistic is the
distribution of values taken by the statistic in all
possible samples of the same size from the same
population.
* Parameter – number that describes a characteristic of
a population (like a mean or proportion) – use Greek
letters to denote.
* Statistic – number that describes a characteristic of a
sample, often used to estimate to a parameter.
* Sampling Variability – If you repeatedly sample from a
population, the statistic you measure varies from
sample to sample even though the parameter is
constant.
* Population Distribution – a graph that shows how a
characteristic is represented in a population (for
example, 30% red, 70% blue) Maybe a bar graph.
* Usually we don’t know the population parameter
distribution, so we take a sample. The sample
might be 25% red and 75% blue. We could graph
this as a sample distribution. Bar Graph.
* We could then graph the proportion of red on a
number line, take a new sample, graph that, and
do that over and over until we’ve gotten every
possible sample. That dotplot or histogram would
be the sampling distribution for the proportion of
red in the population.
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* A statistic is an unbiased estimator of a parameter
if the mean of its sampling distribution if equal to
the true value of the parameter being estimated.
* Means and proportions of good samples are
unbiased estimators, so making a sampling
distribution can give us great information about the
true population parameter.
* This leads us to many powerful inference
techniques for estimating and evaluating evidence
against hypothesized population parameters.
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*The variability of a sampling distribution is
described by the spread of its sampling
distribution. To decrease the spread, use a
larger sample size in building the sampling
distribution.
*The ideal is low variability, low bias – i.e.
large sample size, random sampling of an
unbiased estimator.
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* The sampling distribution for a proportion can
be modeled by N(p, sqrt(pq/n))
* IF the following conditions are met:
* 10%, Success/Fail (see Ch 17 show)
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*According to the Central Limit
Theorem (CLT), the sampling
distribution for a mean can be
modeled by
* N(µ,σ/sqrt(n))
*If the following conditions are met:
*Random sampling, independence,
10%
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* The Standard Deviation of a sampling distribution
for a proportion is sqrt(pq/n) but often we do not
know p and q – population parameters. So,
* We estimate them with p-hat and q-hat, and this
makes our standard deviation an estimate, too.
Since it is an estimate it is called a standard error.
* The Standard Deviation of a sampling distribution
for a mean is sigma/sqrt(n), but we rarely know
sigma, so we estimate it with Sx, and then the
standard deviation estimate is called a standard
error.
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