Sampling distribution of the mean from a normal distribution

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Sampling distribution of the mean from a normal
distribution

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Populations and samples
Statistics are often based on samples of
populations. A common example would be during an
election an opinion poll is often conducted to
find how the general population will vote.
By taking a sample we can find a sample mean and
a sample standard deviation. This is used to
provide us with unbiased estimators for the
population mean and the population standard
deviation.
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Populations and samples - notation
Sample sample mean and sample variance

Population population mean
population variance
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Sampling exercise
Go through the practical exercise on sampling
using Excel - sampling resource sheet 1.
Did you notice a connection between the sample
mean and the population mean?
Did you notice a connection between the sample
variance and the population variance?
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Populations and samples - notation
An unbiased estimate of the population mean, ,
can be found by taking the sample mean, .
An unbiased estimate of the population variance,
, can be found by taking the sample variance,
,and multiplying by n.
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Unbiased estimators of normal distribution
samples - proofs
Mean
Variance
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The standard error of the mean
Very often the term standard error or standard
deviation of the sample mean is used. This is
simply the standard deviation of the sample or S.
Standard error
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Using the sample to make predictions about the
population.
Boys masses are known to be normally distributed.
A sample of 25 boys was taken from a population
and the sample mean and variance were calculated
as being 72 kg and 4 kg respectively.
a) Calculate unbiased estimators for the
population. b) Calculate the probability that a
boy picked at random from the population has a
mass of between 60 kg and 80 kg.
a) mean 72 kg, variance 25x4100 kg
b) PopN(72,100). Draw a diagram to show the
probability required. Note that you must use the
population mean and variance for your
calculations.
Using a GDC (60,80,10,72)0.67 or by using the
z-figures in the diagram above gives the same
probability.