Download Chapter 8

Sampling Methods and
the Central Limit Theorem
Chapter 8
McGraw-Hill/Irwin
©The McGraw-Hill Companies, Inc. 2008
GOALS
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Understand why using sample is the only
feasible way to learn about a population.
Describe methods to select a sample.
Define and construct a sampling
distribution of the sample mean.
Explain the central limit theorem.
Use the central limit theorem to find
probabilities of selecting possible sample
means from a specified population.
Why Sample?
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The physical impossibility of checking all
items in the population.
The cost of studying all the items in a
population.
Contacting the whole population would
often be time-consuming.
The destructive nature of certain tests.
Sample results are usually adequate.
Methods of Probability Sampling
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Simple Random Sample: A sample formulated
so that each item or person in the population
has the same chance of being included.
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Systematic Random Sampling: The items or
individuals of the population are arranged in
some order. A random starting point is
selected and then every kth member of the
population is selected for the sample.
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k = population size/sample size
Methods of Probability Sampling
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Stratified Random Sampling: A population is
first divided into subgroups, called strata, and
a sample is selected from each stratum.
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Cluster Sampling: A population is first divided
into clusters using a natural criterion, like
geographic areas. Then some clusters are
randomly selected, and a sample is selected
from the selected clusters (primary units).
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To represent diverse groups.
To reduce the cost of sampling.
Sampling Error
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The sampling error is the difference between a
sample statistic and the corresponding population
parameter.
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Sample is used to estimate population parameter.
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Use sample mean (s.d) for population mean (s.d.).
It is unlikely that sample mean (s.d.) is equal to population
mean (s.d.).
(Eg.)  = 3 (N=200) for a population.
The first sample (n=20) has mean= 3.5 (sampling error .5)
The second sample (n=20) has mean= 3.2 ( ----- .2)
The third sample (n=20) has mean= 2.6 (sampling error -.4)
Sampling Distribution of the
Sample Means
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The sample mean itself is a random
variable having a certain probability
distribution.
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The sampling distribution of the
sample mean is
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Consider all possible samples of the given size,
getting the mean for each sample.
a probability distribution of all possible sample
means of a given sample size.
Sampling Distribution of the
Sample Means - Example
A company has seven employees (the population). The hourly
earnings of each employee are given in the table below.
1. What is the population mean?
2. What is the sampling distribution of the sample mean of samples of size 2?
3. What is the mean of the sampling distribution?
4. What observations can be made about the population and the sampling
distribution?
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Sampling Distribution of the
Sample Means - Example
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Sampling Distribution of the
Sample Means - Example
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Sampling Distribution of the
Sample Means - Example
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Sampling Distribution of the Sample
Means - Example
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From the above comparison, we can make
following observations.
1) The mean of the sample means is equal to the
mean of the population.
2) The dispersion of the sampling distribution of
sample means is narrower than the population
distribution.
3) The sampling distribution of sample means tends
to become bell-shaped, and to approximate a normal
distribution.
Central Limit Theorem
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For a population with a mean μ and a variance
σ2, the sampling distribution of the means of all
possible samples of size n will be
approximately normally distributed.
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The mean of the sampling distribution equal to
μ and the variance (of the sampling distrubition
of the sample mean) equal to σ2/n.
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This approximation improves with larger samples.
The standard error of the sample means equals to
σ/n1/2
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Central Limit Theorem (example)
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The example in the box (pp. 276-279)
illustrates the central limit theorem
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sample means from a population that is not
normally distributed will converge to normal
distribution.
Chart 8-3 the population distribution (non-normal).
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The years of employment for a total of 40 employees.
Chart 8-4 histogram of the means for 25 samples
of 5 employees.
Chart 8-5 histogram of the means for 25 samples
of 20 employees
Using the Sampling
Distribution of the Sample Mean (σ Known)
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If a population follows the normal distribution,
the sampling distribution of the sample mean
will also follow the normal distribution.
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Regardless of the sample size.
If the distribution of a population is not known
(or it follows the non-normal distribution), the
sampling distribution of the sample means
from at least 30 observations will follow the
normal distribution. (by the CLT).
Using the Sampling
Distribution of the Sample Mean (σ Known)
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We can convert any normal distribution to
standard normal distribution.
To determine the probability a sample mean
falls within a particular region, we use:
z
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X 

n
Using the Sampling
Distribution of the Sample Mean (σ Unknown)
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Suppose the variance (the standard
deviation) of the population is not known.
To determine the probability a sample mean
falls within a particular region, use:
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Topic of the next chapter.
X 
t
s n
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Using the Sampling Distribution of the Sample
Mean (σ Known) - Example
A Cola company maintains records of the amount of cola in a
bottle. The actual amount of cola in each bottle is critical, but
varies a little from one bottle to the next. The company does not
wish to underfill or overfill the bottles. Its records indicate that
the amount of cola follows the normal probability distribution.
The mean amount per bottle is 31.2 ounces and the population
standard deviation is 0.4 ounces. This morning a technician
randomly selected 16 bottles from the filling line. The (sample)
mean amount of cola contained in the bottles was 31.38 ounces.
Is this an unlikely result? To put it another way, is the sampling
error of 0.18 ounces unusual?
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Using the Sampling Distribution of the Sample
Mean (σ Known) - Example
Step 1: Find the z-values corresponding to the
sample mean of 31.38
X   31.38  32.20
z
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 1.80
 n
$0.2 16
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Using the Sampling Distribution of the Sample
Mean (σ Known) - Example
Step 2: Find the probability of observing a Z equal
to or greater than 1.80
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Using the Sampling Distribution of the Sample
Mean (σ Known) - Example
What do we conclude?
It is unlikely, less than a 4 percent chance, we
could select a sample of 16 observations
from a normal population with a mean of 31.2
ounces and a population standard deviation
of 0.4 ounces and find the sample mean
equal to or greater than 31.38 ounces.
We conclude the process is putting too much
cola in the bottles (They have to check the
bottling process)..
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End of Chapter 8
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