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THE DISTRIBUTION OF SAMPLE MEANS
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Chapter 7
THE DISTRIBUTION
OF SAMPLE MEANS
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THE DISTRIBUTION OF SAMPLE MEANS
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Imagine an urn filled with balls…
 Two-third of the balls are one color, and the
remaining one-third are a second color
 One individual select 5 balls from the urn and
finds that 4 are red and 1 is white
 Another individual select 20 balls and finds
that 12 are red and 8 are white
 Which of these two individuals should feel
more confident that the urn contains two third
red balls and one-third white balls than the
opposite?
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THE DISTRIBUTION OF SAMPLE MEANS
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The CORRECT ANSWER
 The larger sample gives a much stronger
justification for concluding that the balls in the
urn predominantly red
 With a small number, you risk obtaining an
unrepresentative sample
 The larger sample is much more likely to provide
an accurate representation of the population
 This is an example of the law of large number
which states that large samples will be
representative of the population from which they
are selected
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THE DISTRIBUTION OF SAMPLE MEANS
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OVERVIEW
 Whenever a score is selected from a
population, you should be able to compute a
z-score
 And, if the population is normal, you should
be able to determine the probability value for
obtaining any individual score
 In a normal distribution, a z-score of +2.00
correspond to an extreme score out in the tail
of the distribution, and a score at least large
has a probability of only p = .0228
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THE DISTRIBUTION OF SAMPLE MEANS
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OVERVIEW
 In this chapter we will extend the concepts of
z-scores and probability to cover situation
with larger samples
 We will introduce a procedure for
transforming a sample mean into a z-score
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THE DISTRIBUTION OF SAMPLE MEANS
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THE DISTRIBUTION OF SAMPLE MEANS
 Two separate samples probably will be
different even though they are taken from the
same population
 The sample will have different individual,
different scores, different means, and so on
 The distribution of sample means is the
collection of sample means for all the possible
random samples of a particular size (n) that
can be obtained from a population
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THE DISTRIBUTION OF SAMPLE MEANS
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COMBINATION
n!
nCr =
r! (n-r)!
 Consider a population that consist of 5 scores:
3, 4, 5, 6, and 7
 Mean population = ?
 Construct the distribution of sample means for
n = 1, n = 2, n = 3, n = 4, n = 5
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THE DISTRIBUTION OF SAMPLE MEANS
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SAMPLING DISTRIBUTION
 … is a distribution of statistics obtained by selecting
all the possible samples of a specific size from a
population
CENTRAL LIMIT THEOREM
 For any population with mean μ and standard
deviation σ, the distribution of sample means for
sample size n will have a mean of μ and a standard
deviation of σ/√n and will approach a normal
distribution as n approaches infinity
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THE DISTRIBUTION OF SAMPLE MEANS
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The STANDARD ERROR OF MEAN
 The value we will be working with is the
standard deviation for the distribution of
sample means, and it called the σM
 Remember the sampling error
 There typically will be some error between
the sample and the population
 The σM measures exactly how much
difference should be expected on average
between sample mean M and the population
mean μ
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THE DISTRIBUTION OF SAMPLE MEANS
The MAGNITUDE of THE σM
 Determined by two factors:
○ The size of the sample, and
○ The standard deviation of the population from
which the sample is selected
M 
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THE DISTRIBUTION OF SAMPLE MEANS
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LEARNING CHECK
 A population of scores is normal with μ = 100
and σ = 15
○ Describe the distribution of sample means for
samples size n = 25 and n =100
 Under what circumstances will the
distribution of samples means be a normal
shaped distribution?
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THE DISTRIBUTION OF SAMPLE MEANS
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PROBABILITY AND THE DISTRIBUTION
OF SAMPLE MEANS
 The primary use of the standard distribution
of sample means is to find the probability
associated with any specific sample
 Because the distribution of sample means
present the entire set of all possible Ms, we can
use proportions of this distribution to
determine probabilities
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THE DISTRIBUTION OF SAMPLE MEANS
EXAMPLE
 The population of scores on the SAT forms a
normal distribution with μ = 500 and σ = 100.
If you take a random sample of n = 16
students, what is the probability that sample
mean will be greater that M = 540?
σM =
σ
√n
= 25
M-μ
z= σ
M
= 1.6
z = 1.6  Area C  p = .0548
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THE DISTRIBUTION OF SAMPLE MEANS
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LEARNING CHECK
 The population of scores on the SAT forms a
normal distribution with μ = 500 and σ = 100.
We are going to determine the exact range of
values that is expected for sample mean 95%
of the time for sample of n = 25 students
See Example 7.3 on Gravetter’s book page 207
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