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Sample Variability
Consider the small population of integers {0, 2, 4, 6, 8}
It is clear that the mean, μ = 4. Suppose we did not know the
population mean and wanted to estimate it with a sample mean
with sample size 2. (We will use sampling with replacement)
We take one sample and get sample mean, ū1 = (0+2)/2 = 1 and
take another sample and get a sample mean ū2 = (4+6)/2 = 5.
Why are these sample means different?
Are they good estimates of the true mean of the population?
What is the probability that we take a random sample and get a
sample mean that would exactly equal the true mean of the
population?
Section 7.1, Page 137
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Sampling Distribution
Each of these samples has a sample mean, ū.
These sample means respectively are as
follows:
P(ū = 1) = 2/25 = .08
P(ū = 4) = 5/25 = .20
Section 7.1, Page 138
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Sampling Distribution
Shape is normal
Mean of the sampling
distribution = 4, the mean
of the population
Section 7.1, Page 138
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Sampling Distributions and
Central Limit Theorem
Alternate notation:
SE(x )

Sample sizes ≥ 30 will assure
a normal distribution.
Section 7.2, Page 141
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Central Limit Theorem
Section 7.2, Page 144
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Central Limit Theorem
Section 7.2, Page 145
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
Calculating Probabilities for the
Mean
Kindergarten children have heights that are approximately
normally distributed about a mean of 39 inches and a
standard deviation of 2 inches. A random sample of 25 is
taken. What is the probability that the sample mean is
between 38.5 and 40 inches?
P(38.5 < sample mean <40) =
NORMDIST 1
LOWER BOUND = 38.5
UPPER BOUND = 40
MEAN =39
SE(x )  2/ 25  0.4
ANSWER: 0.8881

Section 7.3, Page 147
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
Calculating Middle 90%
Kindergarten children have heights that are approximately
normally distributed about a mean of 39 inches and a
standard deviation of 2 inches. A random sample of 25 is
taken. Find the interval that includes the middle 90% of all
sample means for the sample of kindergarteners.
Sampling Distribution
x 
2
25

ux  39
NORMDIST 2
AREA FROM LEFT =0.05
MEAN = 39
NORMDIST 2
AREA FROM LEFT = .95
MEAN = 39
ANSWER: 38.3421
ANSWER: 39.6579
SE(x )  2/ 25  0.4
SE(x )  2/ 25  0.4
The interval (38.3 inches, 39.7 inches) contains the
 means. If we choose a
middle 90% of all sample
random sample, there is a 90% probability that it will
be in the interval.
Section 7.3, Page 147
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Problems
Problems, Page 149
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Problems
Problems, Page 150
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Problems
Problems, Page 151
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