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Sample Size
Determination
Chapter
Eleven
Copyright © 2006
John Wiley & Sons, Inc.
Learning Objectives
1. To learn the financial and statistical issues in
the determination of sample size
2. To discover methods for determining sample
size
3. To gain an appreciation of a normal
distribution
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Learning Objectives
4. To understand population, sample, and
sampling distributions.
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
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 on a liner basis; however, the
sampling error decreases at a rate equal to the square root
of the relative increase in sample size.
– Before trying to determine the size of the sample, the
confidence intervals need to be decided.
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Determining Sample Size
for Probability Samples
The financial and statistical issues in
the determination of sample size
• Budget Available
– Sample Size—a project is often determined by the
available budget
– Alternative Data Collection Approaches
• budget constraints force the researcher to explore and
consider the value of information in relation to its cost
• Rules of Thumb
– Desired sampling error
– Past experience
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- Similar Studies
- A gut feeling
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Determining Sample Size
for Probability Samples
The financial and statistical issues in
the determination of sample size
• Number of Subgroups To Be Analyzed
– Subgroups
– Samples size
– Minimum needs
• Traditional Statistical Methods
– 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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The Normal Distribution
To gain an appreciation
of a normal distribution
• General Properties for the Normal Distribution
– Crucial to Classical Statistical Inference
• Reasons For Its Importance
– Many variables have probability distributions that are close
to the normal distribution
– Central Limit Theorem—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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The Normal Distribution
•
To gain an appreciation
of a normal distribution
Important Characteristics of the Normal Distribution
1. The 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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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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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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Population and Sample
Distributions
To understand population, sample,
and sampling distributions.
• Purpose of Conducting a Survey
– To make inferences
– Sample
– Population Distribution
• A frequency distribution of all the elements of a
population. It has a mean (µ)
– Sample Distribution
• A frequency distribution of all the elements of an
individual sample.
– The mean x
– The standard divergence—S
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Sampling Distributions
Of The Mean
To understand population, sample,
and sampling distributions.
• If the samples are sufficiently large and random,
the resulting distribution of sample means will
approximate a normal distribution.
• 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  and a standard deviation equal
to:
sx
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=

√ n
12
Sampling Distributions
Of The Mean
To understand population, sample,
and sampling distributions.
• Basic Concepts
– Characteristics of a Large Simple Random
Sample
•
•
•
•
30 or more observations
Distribution is a normal distribution
Distribution has a mean equal to the population mean
Distribution has a standard deviation, referred to as the
standard error of the mean, equal to the population standard
deviation divided by the square root of the sample
x
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=

√ n
13
Sampling Distributions
Of The Mean
To understand population, sample,
and sampling distributions.
• Making Inferences on the Basis of a Single Sample
– A 68.26% probability that a random sample will produce an
estimate of the population mean within one standard error
(±) of the true mean
– 95.44 % probability that a random sample will produce an
estimate of the population mean within two standard error
(±) of the true mean
– 99.74 % probability that a random sample will produce an
estimate of the population mean within three standard error
(±) of the true mean
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Sampling Distributions
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.
• Sampling Error
– The distance between the x and Χ
• Interval Estimate
– The likelihood that a population value falls within a certain
range.
• Confidence Interval
– The interval that in all probability, includes the true
population value
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Sampling Distributions
Of The Mean
To distinguish between point
and interval estimates.
• Method of Deriving Interval Estimates
– Draw a random sample from the population
interest and calculating the mean of that sample
– There is a 68.26% probability that the sample
mean lies within one σ x
– Below is the statement symbolically
x
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1 x <  < x + 1 x
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Sampling Distributions
Of The Mean
To recognize problems involving
sampling means and proportions.
• Interest in Estimating Proportions or Percentages
– Rather than or in addition to estimating means.
Examples
• The percentage of the population that is aware of a
particular
• The percentage of the population that accesses the
Internet one or more times in an average week
• The percentage of the population that has visited a fastfood restaurant four or more times in the past 30 days
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Sampling Distributions
Of The Mean
•
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 computed as:
Sp
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=
√ P (1-P)
n
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Determining
Sample Size
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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Determining
Sample Size
To recognize problems involving
sampling means and proportions.
• Information needed to compute the sample size
• Allowable sampling error
• Population standard deviation
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Determining
Sample Size
•
To recognize problems involving
sampling means and proportions.
Four Methods Used to Deal with the Estimate
of the Population Standard Deviation
1.
2.
3.
4.
Use results from a prior survey
Conduct a pilot survey
Use secondary date
Use judgment
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Determining
Sample Size
To recognize problems involving
sampling means and proportions.
• Problems Involving Proportions
– If there is no basis for estimating P, the researcher
can make the most pessimistic case regarding the
value of P
– Given the values of Z and E, what value of P will
require the largest possible sample
– A value of .50 will make the value of the
expression P(1-P) larger than any possible value of
P.
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Determining
Sample Size
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.
x
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=

√ n
√
N-n
N-1
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Determining
Sample Size
To recognize problems involving
sampling means and proportions.
– Finite Population Correction (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)
n' =
nN
N + n -1
where:
n' = revised sample size
n = original sample size
N = population size
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Statistical Power
To recognize problems involving
sampling means and proportions.
• Types of Errors
– Type I Error
• The error of concluding that there is a difference
when there is no difference
– The formulas in this chapter focus on this type of error
– Type II Error
• The error of saying that there is no difference when
there is a difference
– Statistical Power
• The probability of not making a Type II error
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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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The End
Copyright © 2006 John Wiley & Sons, Inc.
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