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Part 4
Sampling and
Data Collection
DETERMINATION
OF SAMPLE SIZE:
A REVIEW OF
STATISTICAL THEORY
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
Chapter
15
LEARNING OBJECTIVES
What you will learn in this chapter
1. To discuss the purpose of inferential statistics by
explaining the difference between population
parameters and sample statistics
2. To make data usable by organizing and summarizing
them into frequency distributions, proportions, and
measurements of central tendency
3. To identify and calculate the various measures of
central tendency and dispersion
4. To identify the characteristics of the normal distribution
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–1
LEARNING OBJECTIVES (cont’d)
What you will learn in this chapter
5. To distinguish among population, sample, and sampling
distributions and to identify the mean and standard
deviation of each distribution
6. To compute confidence interval estimates
7. To understand the factors required for specifying
sample size
8. To estimate the sample size for a simple random
sample when the characteristic of interest is a mean
and when it is a proportion
9. To understand which nonstatistical considerations
influence the determination of sample size
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–2
Reviewing Basic Terminology
• Descriptive and Inferential Statistics
Descriptive statistics describe characteristics of
populations or samples
Inferential statistics is used to make inferences about
a whole population from a sample
• Sample Statistics and Population Parameters
Sample statistics
 Variables in a sample or measures computed from sample
data
Population parameters
 Variables in a population or measured characteristics of the
population
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–3
Making Data Usable
• Frequency Distributions
Frequency distribution
 A set of data organized by summarizing the number of times
a particular value of a variable occurs
Percentage distribution
 A frequency distribution organized into a table (or graph)
that summarizes percentage values associated with
particular values of a variable
Probability
 The long-run relative frequency with which an event will
occur
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–4
Making Data Usable (cont’d)
• Proportions
The percentage of elements that meet some criterion
• Measures of Central Tendency
The mean
 A measure of central tendency; the arithmetic average
The median
 A measure of central tendency that is the midpoint; the
value below which half the values in a distribution fall
The mode
 A measure of central tendency; the value that occurs most
often
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–5
Making Data Usable (cont’d)
• Measures of Dispersion
The range
 The distance between the smallest and the largest values of
a frequency distribution
Deviation scores
 A method of calculating how far any observation is from the
mean
 Example:
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–6
Making Data Usable (cont’d)
• Measures of Dispersion (cont’d)
Why use the standard deviation?
 Variance: A measure of variability or dispersion. Its square
root is the standard deviation
 Standard deviation: A quantitative index of a distribution’s
spread, or variability; the square root of the variance for a
distribution
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–7
The Normal Distribution
• Normal Distribution
A symmetrical, bell-shaped distribution that
describes the expected probability distribution of
many chance occurrences
99% of its values are within ± 3 standard deviations
from its mean
• Standardized Normal Distribution
A purely theoretical probability distribution that
reflects a specific normal curve for the standardized
value, Z
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–8
Population Distribution, Sample Distribution,
and Sampling Distribution
• Population Distribution
A frequency distribution of the elements of a
population
• Sample Distribution
A frequency distribution of a sample
• Sampling Distribution
A theoretical probability distribution of sample means
for all possible samples of a certain size drawn from
a particular population
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–11
Population Distribution, Sample Distribution,
and Sampling Distribution (cont’d)
• Standard Error of the Mean
The standard deviation of the sampling distribution
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–12
Central-Limit Theorem
• Central-Limit Theorem
The theory that, as sample size increases, the
distribution of sample means of size n, randomly
selected, approaches a normal distribution
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–13
Estimation of Parameters
• Point Estimates
An estimate of the population mean in the form of a
single value, usually the sample mean
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–14
Estimation of Parameters (cont’d)
• Confidence Intervals
Confidence interval estimate
 A specified range of numbers within which a population
mean is expected to lie; an estimate of the population mean
based on the knowledge that it will be equal to the sample
mean plus or minus a small sampling error
Confidence level
 A percentage or decimal value that tells how confident a
researcher can be about being correct. It states the long-run
percentage of confidence intervals that will include the true
population mean
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–15
Sample Size
• Random Error and Sample Size
Random sampling error varies with samples of
different sizes
Increasing the sample size decreases the width of
the confidence interval at a given confidence level
• Factors in Determining Sample Size for
Questions Involving Means
1. The variance, or heterogeneity, of the population
2. The magnitude of acceptable error
3. The confidence level
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–16
Sample Size (cont’d)
• Estimating Sample Size for Questions Involving
Means
Estimate the standard deviation of the population
Make a judgment about the allowable magnitude of
error
Determine a confidence level
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–17
Sample Size (cont’d)
• The Influence of Population Size on Sample
Size
In most cases the size of the population does not
have a major effect on the sample size
The variance of the population has the largest effect
on sample size
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–18
Sample Size (cont’d)
• Factors in Determining Sample Size for
Proportions
When the question involves the estimation of a
proportion, the researcher requires some knowledge
of the logic for determining a confidence interval
around a sample proportion estimation of the
population proportion
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–19
Sample Size (cont’d)
• Calculating Sample Size for Sample
Proportions
In practice, a number of tables have been
constructed for determining sample size
• Determining Sample Size on the Basis of
Judgment
Just as sample units may be selected to suit the
convenience or judgment of the researcher, sample
size may also be determined on the basis of
managerial judgments
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–20
Sample Size (cont’d)
• Determining Sample Size for Stratified and
Other Probability Samples
Stratified sampling involves drawing separate
probability samples within the subgroups to make the
sample more efficient
With a stratified sample the sample variances are
expected to differ by strata
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–21
A Reminder About Statistics
• Learning the terms and symbols defined in this
chapter will provide you with the basics of the
language of statisticians and researchers
• As you learn more about the pragmatic use of
statistics in marketing research, do not forget
these concepts
Copyright © 2008 by Nelson, a division of Thomson Canada Limited
15–22