Sampling
Distributions &
Sample Means
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Why Sample?
• A sample is a small portion of the
population that is designed to give
accurate results about the
population
• Sampling saves a lot of time and
money and is more realistic
Parameter vs. Statistic
• Parameter
– Number that describes the population
• Often don’t know cuz we can’t examine a
population (most of the time)
• Fixed #
• Statistic:
– # that describes the sample.
• Found from taking a sample
• Changes from sample to sample
• Used to estimate unknown parameters
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These are
either ideal
#’s created
by
company or
projected
#’s… Not
KNOWN for
large
populations
!!
Parameter vs. Statistic
• Parameters
–  - population mean
–  - population standard deviation
– p – population proportion
• Statistic:
Mean of Means =
– x-bar – sample mean
– s – sample standard deviation
– p-hat – sample proportion
These #’s are collected from samples.
Samples taken should be the same size. Each
sample can give different results…
What’s the Problem with
Sampling?
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What’s Your Weight?
• Different Samples Give Different
Results
Sampling Variability
• We often use “statistics” to make
inferences about the population
So What’s the Problem?
The value of a statistic varies from sample to
sample when you are taking repeated random
samples.
The concept listed above is called
Sampling Variability.
How do we counteract this problem?
We take a whole bunch o
samples, hence sampling
distributions.
Sampling Distribution
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• Sampling Distribution of a Statistic
– the distribution of values taken by the
statistic in all possible samples of the
same size from the same population
– Ideal pattern that would be displayed if we
looked at all possible samples of a
population
– Used to make inferences about
parameters
– Distribution of all collected sample
means defined by a mean and standard
deviation
Describe a Sampling
Distribution
• Center
– Description of μxbar x-bar or p-hat vs. the
median
• Shape
– “often” normal, but looking for skewness
• Spread
– Talk about range and standard deviation
(std dev is especially strong descriptor for
normal distributions), also look for
outliers
Fighting Sampling
Variability
• Below are 4 keys to fighting Sampling
Variability
– Take a large # of samples from the same
population (all the same size)
• bigger samples are always better!!
– Calculate the sample mean, x-bar, or the
sample proportion, p-hat, for each sample
– Make a histogram of the values of x-bar or phat
– Examine the distribution displayed in the
histogram for shape, center, and spread, as
well as outliers and other deviations
Sampling
Distribution
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Bias of Statistic
• Bias – Difference from Overall Sample
Mean (from Sampling Distribution) and
Unknown Population Mean
– Relation of the statistic to the parameter
– Unbiased Statistic:
• Mean of sampling distribution is equal to the true
value of the parameter being estimated
• μxbar = , p-hat = p
• Variability – difference in values from
sample to sample
– The spread of the sampling distribution
• Larger samples = Less variability(spread)
Bias
vs.
Variability
If I was
a sampling
That’s
the
Good
Shootin’
Practice
distribution, Target
I’d
what
we like to
Pardner!!!
want to be THAT
SEE!!
ONE!!!!
High Variability,
Low Bias
Low Variability,
Low Bias
Low Variability,
High Bias
High Variability,
High Bias
Sampling Distributions
of Means (quantitative)
• μxbar is unbiased • Standard
estimator of µ
• μxbar = µ
• Means are less
variable than
individual
observations
• Means are
MORE normal
Deviation of
means is effected
by the sample
size
x 

n
Variation of Means
• 4 times sample size to ½ std
dev
• Population should be 10 times
size of sample
– Formulas and characteristics
may not be accurate if this
condition is not met
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Let’s
Practice
This From
Our Height
Worksheet
… Smurf
Worksheet
Using Normality to Your
Advantage
• Samples taken from Normal
Distributions are normal…
• Allows us to use the normal curve
to calculate the probability of your
sample occurring
• EX: What is the probability that a
randomly selected student would be
shorter than 63 inches?
• What is the probability that an SRS of
15 students has a mean height less than
63 inches?
Central Limit Theorem
• What if it’s not normal??
• Bigger sample size more normal
distribution
• How big?
Look at your Height Worksheet…
• The moreHow
outcan
of wack,
the bigger
you compare
the shape
the distributions
as you
your samplesof need
to become
increased the sample size?
For your thoughts…
Homework
Problem #’s 12-17,35-38
Bring the Coins Listed
Below and Fill Out Part
1 on the Worksheet
50
2
10
10