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CHAPTER thirteen
Learning Objectives
Sample Size
Determination
Copyright © 2002
South-Western/Thomson Learning
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Learning Objectives
Learning Objectives
1. To learn the financial and statistical issues in
the determination of sample size.
2. To discover the methods for determining
sample size.
3. To gain an appreciation of a normal
distribution.
4. To understand population, sample, and
sampling distribution.
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Learning Objectives
Learning Objectives
5. To distinguish between point and interval
estimates.
6. To recognize problems involving sampling
means and proportions.
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Determining Sample Size
for Probability Samples
Learning Objectives
The financial and statistical
issues in the determination of
sample size.
Financial, Statistical, and Managerial Issues
As a general rule:
The larger the sample, the smaller the sampling error.
Larger samples cost more.
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Methods for Determining
Sample Size
Learning Objectives
The financial and statistical
issues in the determination of
sample size.
Budget Available
Financial constraints challenge the researcher.
Rules of Thumb (how to choose):
• Desired sampling error
• Past experience
• Similar studies
• A gut feeling
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Learning Objectives
Methods for Determining
Sample Size
To discover the methods for
determining sample size.
Number of Subgroups To Be Analyzed
The sample should contain at least 100 respondents in
each major subgroup (e.g., males vs. females).
Traditional Statistical Methods – Need 3 things:
• An estimate of the population standard deviation.
• The acceptable level of sampling error.
• The desired level of confidence that the sample will fall
within a certain range of the true population values.
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Learning Objectives
The Normal Distribution
To gain an appreciation of
a normal distribution.
General Properties
1. Normal distribution is bell-shaped and has only one mode.
2. Symmetrical about the mean
3. Uniquely defined by its mean and standard deviation.
4. The total area is equal to one.
5. The area between any two values of a variable equals the
probability of observing a value in that range when
randomly selecting an observation from the distribution.
6. The area between the mean and a given number of
standard deviations from the mean is the same for all
normal distributions.
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Learning Objectives
The Normal Distribution
To gain an appreciation of
a normal distribution.
Central Limit Theorem
A distribution of a large number of sample means or
sample proportions will approximate a normal distribution,
regardless of the distribution of the population from which
they were drawn
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Learning Objectives
The Normal Distribution
To gain an appreciation of
a normal distribution.
The Standard Normal Distribution
• The same features as any normal distribution.
• The mean is equal to zero
• The standard deviation is equal to one.
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Learning Objectives
The Normal Distribution
To gain an appreciation of
a normal distribution.
value of the variable - mean of the variable
Z=
standard deviation of the variable
where
X-
Z =

X = value of the variable
 = mean of the variable
 = standard deviation of the variable
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Learning Objectives
Population, Sample, and
Sampling Distributions
To understand population, sample,
and sampling distributions.
Population Distribution
A frequency distribution of all the elements of a
population.
Sample Distribution
A frequency distribution of all the elements of an
individual sample.
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Learning Objectives
Population, Sample, and
Sampling Distributions
To understand population, sample,
and sampling distributions.
Sampling Distribution of the Sample Mean
A frequency distribution of the means of many sample
means from a given population
If the samples are sufficiently large and random, the
resulting distribution of sample means will approximate a
normal distribution.
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Learning Objectives
Population, Sample, and
Sampling Distributions
To understand population, sample,
and sampling distributions.
Sampling Distribution of the Sample Mean
The distribution of the means of a large number of random
samples taken from virtually any population approaches a
normal distribution with a mean equal to  (n = sample
size) and a standard deviation equal to:
sx
=

√ n
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Learning Objectives
Population, Sample, and
Sampling Distributions
To understand population, sample,
and sampling distribution.
The Standard Error of the Mean
Applies to the standard deviation of a distribution of
sample means.
x
=

√ n
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Learning Objectives
Sampling Distribution of
the Mean
To understand population, sample,
and sampling distribution.
Sampling Distribution of the Mean: Basic Concepts
1. A normal distribution
2. Mean equal to the population mean.
3. Standard deviation (defined above)
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Learning Objectives
Sampling Distribution of
the Mean
To understand population, sample,
and sampling distribution.
Making Inferences on the Basis of a Single Sample
A 68 percent probability that any one sample from a population
will produce an estimate of the population mean that is within plus
or minus one standard deviation of the population mean.
A 95 percent probability that any one sample from a population
will produce an estimate of the population mean that is within plus
or minus two standard deviations of the population mean.
A 99.7 percent probability that any one sample from a population
will produce an estimate of the population mean that is within plus
or minus three standard deviations of the population mean.
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Learning Objectives
Sampling Distribution of
the Mean
To distinguish between point and
interval estimates.
Point Estimates
Inferences regarding the sampling error associated with a
particular estimate of the population value.
Interval Estimate
Inference regarding the likelihood that a population value will
fall within a certain range. I.e., True population value is equal
to the sample value plus or minus one standard error.
x
1 x <  < x + 1 x
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Learning Objectives
Sampling Distribution of
the Proportion
To recognize problems involving
sampling means and proportions.
Sampling Distribution of the Proportion
A relative frequency distribution of the sample proportions of a
large number of random samples of a given size drawn from a
particular population.
1. Approximates a normal distribution
2. The mean proportion is equal to the population
proportion.
3. Standard error of sampling distribution computed as:
Sp
=
√ P (1-P)
n
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Learning Objectives
Sampling Distribution of
the Proportion
Sp
To recognize problems involving
sampling means and proportions.
=
√ P (1-P)
n
where:
Sp = standard error of sampling distribution
proportion
P = estimate of population proportion
n = sample size
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Learning Objectives
Sample Size Determination
To recognize problems involving
sampling means and proportions.
Problems Involving Means
The formula for calculating the required sample size for
problems that involve the estimation of a mean:
n
=
Z2 2
E2
where:
Z = level of confidence expressed in
standard errors
 = population standard deviation
E = acceptable amount of sampling error
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Learning Objectives
Sample Size Determination
To recognize problems involving
sampling means and proportions.
Problems Involving Means – Example
You are in charge of planning a chili cook-off. You must make
sure that there are plenty of samples for the patrons of the
cook-off. The following standards have been set: a confidence
level of 99 percent and an error of less than .4 ounces per
cooking team. Last year’s cook-off had a standard deviation in
amount of chili cooked of 3 ounces. What is the necessary
sample size?
Z = 3 (from table 13.1)
=3
E = .4
Plug it all in, and n = 506
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Learning Objectives
Sample Size Determination
To recognize problems involving
sampling means and proportions.
Problems Involving Proportions
n
=
Z2 [P(1-P)]
E2
where:
Z = level of confidence expressed in standard errors
P = Proportion
E = acceptable amount of sampling error
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Learning Objectives
Sample Size Determination
To recognize problems involving
sampling means and proportions.
Problems Involving Proportions: Example
n
=
Z2 [P(1-P)]
E2
Assume previous fast food research has shown that 80 percent of the
consumers like curly french fries. The researcher wishes to have a
standard error of 6 percent or less, and be 95 percent confident of an
estimate to be made about curly french fry consumption from a survey.
What sample size should be used for a simple random sample?
Z = 2 (see table 13.1)
P = .8
E = .06
Plug it all in, and n = 178
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Learning Objectives
Sample Size Determination
To recognize problems involving
sampling means and proportions.
Population Size and Sample Size
Make an adjustment in the sample size if the sample size is
more than 5 percent of the size of the total population.
Finite Population Correction Factor (FPC)
An adjustment in cases where the sample is
expected to be equal to 5 percent or more of the total
population. (N-n) / (N-1)
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Learning Objectives
Sample Size Determination
To recognize problems involving
sampling means and proportions.
Adjusting for a sample that is 5 percent or more of the
population and dropping the independence assumption:
x
=

√ n
√
N-n
N-1
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Learning Objectives
Sample Size Determination
To recognize problems involving
sampling means and proportions.
Reducing the required sample size using the Finite
Population Correction
n' =
nN
N + n -1
where:
n' = revised sample size
n = original sample size
N = population size
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Learning Objectives
Sample Size Determination
To recognize problems involving
sampling means and proportions.
Determining Sample Size for Stratified and Cluster
Sample
• Beyond the scope of this text.
Determining How Many Sample Units You Need
• Don’t want to pay for more numbers than needed
• Don’t want to run out of numbers.
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Learning Objectives
Statistical Power
To recognize problems involving
sampling means and proportions.
Statistical Power
The probability of not making a Type II error.
Type I Error
The error of concluding that there is a difference when
there is not a difference.
Type II Error
The error of saying that there is no difference when
there actually is a difference.
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Learning Objectives
SUMMARY
• Determining Sample Size for Probability Samples
• Methods for Determining Sample Size
• The Normal Distribution
• Population, Sample, and Sampling Distributions
• Sampling Distribution of the Mean
• Sampling Distribution of the Proportion
• Sample Size Determination
• Statistical Power
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Learning Objectives
The End
Copyright © 2002 South-Western/Thomson Learning
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