Transcript The Distribution of Sample Means

The Distribution of Sample
Means
The Distribution of Sample Means
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Definition: the set of means from all
the possible random samples of a
specific size (n) selected from a
specific population
Theoretical distribution
Central Limit Theorem
1.
The mean of the distribution of sample means is
called the Expected Value of M
2. The standard deviation of the distribution of sample
means is called the Standard Error of M
M 
3.

n
The shape of the distribution of sample means tends
to be normal.
Example
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Imagine a population that is normally
distributed with µ=110 and σ=24
If we take a sample from this
population, how accurately would the
sample mean represent the population
mean?
Probability and Sample Means
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Because the distribution of sample
means tends to be normal, the zscore value obtained for a sample
mean can be used with the unit
normal table to obtain probabilities.
z-Scores and Location within the
Distribution of Sample Means
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Within the distribution of sample
means, the location of each sample
mean can be specified by a z-score,
M–μ
z = ─────
σM
Example
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GRE quantitative scores are considered
to be normally distributed with a  =
500 and  = 100.
An exceptional group of 16 graduate
school applicants had a mean GRE
quantitative score of 710.
What is the probability of randomly
selecting 16 graduate school applicants
with an even greater mean GRE
quantitative score?
p (> 710 ) = ?
Example 2
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Scores on a test form a normal
distribution with µ=70 and σ=12.
With a sample size of n=16.
What is the probability of obtaining
a sample of at least 75?
70
0
75