Download Sample Means

Sample Means
Parameters
 The mean and standard deviation of a population are
parameters.
 Mu represents the population mean.
 Sigma represents the population standard deviation.
 Example: IQ test scores are normally distributed with a
mean (mu) of 100 and a standard deviation (sigma) of 15.
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Statistics
 Suppose that x-bar is the mean of an SRS of size n
drawn from a large population having mean mu and
standard deviation sigma.
 The mean of the sampling distribution of x-bar is 
and its standard deviation is  adjusted by the square
root of the sample size:
x 

n
Notes on
x
and  x
 The sample mean x-bar is an unbiased estimator of the
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population mean mu.
The values of x-bar are less spread out for larger
samples.
The standard deviation decreases at a rate of n . Thus,
you must take a sample four times as large to cut the
standard deviation in half.
The population must be at least ten times the sample
size to use the formula sigma divided by radical n.
These rules apply no matter what shape the
population has.
Sampling Distribution of a Sample
Mean
 If you draw an SRS of any size from a normally
distributed population, the sampling distribution of
the sample means will also be normally distributed.
Example
Central Limit Theorem
 If a sample of size n is large enough, a population with
mean mu and standard deviation sigma will give a
sampling distribution of means which is approximately
normal.
 This is true no matter the shape of the population
distribution.
 Basic rule: if n is greater than or equal to 30, then the
Central Limit Theorem applies.
Example
Law of Large Numbers Revisited
 If you draw observations from any population with a
finite mean mu, the mean x-bar of the observed values
gets closer and closer to mu as the number of trials
approaches the maximum.
 In the long-run, the more samples of a given size
which are taken, the closer the mean of the sampling
distribution approaches the true mean of the
population and stays there.
Homework
 Worksheet