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Chapter 6
Introduction to Sampling
Distributions
©
Chapter 6 - Chapter Outcomes
After studying the material in this chapter,
you should be able to:
• Understand the concept of sampling
error.
• Determine the mean and standard
deviation for the sampling distribution
of the sample mean.
Chapter 6 - Chapter Outcomes
(continued)
After studying the material in this chapter, you
should be able to:
• Determine the mean and standard
deviation for the sampling distribution
of the sample proportion.
• Understand the importance of the
Central Limit Theorem.
• Apply the sampling distributions for
both the mean and proportion.
Sampling Error
SAMPLING ERROR-SINGLE MEAN
The difference between a value (a statistic) computed
from a sample and the corresponding value (a parameter)
computed from a population.
Where:
Sampling Error  x -
x  Sample mean
  Population mean
Sampling Error
-Parameters v. Statistics• A parameter is a measure computed
from the entire population
• A statistic is a measure computed
from a sample that has been selected
from a population.
Sampling Error
POPULATION MEAN
x


Where:
N
 = Population mean
x = Values in the population
N = Population size
Sampling Error
(Example 6.1)
If  = 158,972 square feet and a sample of n = 5
shopping centers yields x = 155,072 square feet,
then the sampling error would be:
x    155,072  158,972  3,900 square feet
Sampling Errors
Useful Fundamental Statistical Concepts:
• The size of the sampling error depends on
which sample is taken.
• The sampling error may be positive or
negative.
• There is potentially a different value for
each possible sample mean.
Sampling Error
A simple random sample is a
sample selected in such a manner
that each possible sample of a
given size has an equal chance of
being selected.
Sampling Error
SAMPLE MEAN
x

x
Where:
n
x = Sample mean
x = Sample value selected from
the population
n = Sample size
Sampling Errors
POPULATION PROPORTION
Where:
x

N
 = Population proportion
x = Number of items having the attribute
N = Population size
Sampling Errors
SAMPLE PROPORTION
Where:
x
p
n
p = Sample proportion
x = Number of items in the
sample having the attribute
n = sample size
Sampling Error
SINGLE PROPORTION SAMPLING ERROR
Sampling Error  ( p - )
Where:
p  Sample proportion
  Population proportion
Sampling Distributions
A sampling distribution is a
distribution of the possible values
of a statistic for a given size sample
selected from a population.
Sampling Distribution of the
Mean
THEOREM 6-1
If a population is normally distributed with a
mean  and a standard deviation , the sampling
distribution of the sample mean x is also
normally distributed with a mean equal to the
population mean (  x   ) and a standard
deviation equal to the population standard
deviation divided by the square-root of the

sample size  x 
.
n
Sampling Distribution of the
Mean
THEOREM 6-2: THE CENTRAL LIMIT THEREOM
For random samples of n observations taken from a
population with mean  and standard deviation ,
regardless of the population’s distribution, provided
the sample size is sufficiently large, the distribution
of the sample mean x , will be normal with a
mean equal to the population mean (  x   )
.
Further, the standard deviation will equal the
population standard deviation divided by the

square-root of the sample size  x 
.
n
The larger the sample size, the better the
approximation to the normal distribution.
Sampling Distribution of the
Mean
z-VALUE FOR SAMPLING DISTRIBUTION OF
z
where:
(x  )

n
= Sample mean
 = Population mean
 = Population standard deviation
n = Sample size
x
x
Example of Calculation
z-Value for the Sample Mean
(Example 6-5)
What is the probability that a sample of 100 automobile
insurance claim files will yield an average claim of
$4,527.77 or less if the average claim for the population is
$4,560 with standard deviation of $600?
z
(x  )

n
(4,527.77  4,560)  32.23


 0.537
600
60
100
P( z  0.537)  0.5000  0.2054  0.2946
Sampling Distribution of a
Proportion
SAMPLING DISTRIBUTION OF p
Mean   p  
and
Standard Error   p 
where:

 (1   )
n
= Population proportion
p = Sample proportion
n = Sample size
Sampling Distribution of the
Mean
z-VALUE FOR PROPORTIONS
z
where:
( p  )
p
z = Number of standard errors p is from
p = Sample proportion
 p = Standard error of the sampling
distribution
 p    Mean of sample proportions

Example of Calculation
z-Value for Proportion
(Example 6-6)
What is the probability that a sample of 500 units will
contain 18% or more broken items given that observation
over time has shown 15% of shipments damaged?
z
( p  )
p

0.18  0.15
 1.88
(0.15)(0.85)
500
P( p  1.88)  0.5000  0.4699  0.0301
Key Terms
• Central Limit Theorem
• Finite Population
Correction Factor
• Parameter
• Population Proportion
• Sample Proportion
• Sampling Distribution
• Sampling Error
• Simple Random Sample
• Statistic
• Theorem 6-1