Lecture Marketing research - Chapter 11: Sample size determination

CHAPTER Eleven

Learning Objectives

Sample Size
Determination

Copyright © 2004
John Wiley & Sons, Inc.


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.


Learning Objectives

Learning Objectives

5. To distinguish between point and interval
estimates.
6. To recognize problems involving sampling

means and proportions.


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; 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.


Methods for Determining
Sample Size

Learning Objectives
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

- Similar Studies

- Past experience

- A gut feeling


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.
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.


Learning Objectives

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


Learning Objectives

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


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.


Learning Objectives
To gain an appreciation of
a normal distribution.

The Normal Distribution

value of the variable - mean of the variable

Z=
standard deviation of the variable

Z =

X-

where

X = value of the variable
= mean of the variable
= standard deviation of the variable


Learning Objectives

Sampling Distributions
Of The Mean

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.
Sampling Distribution of the Sample Mean
A frequency distribution of the means of many sample
means from a given population


Learning Objectives

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

=

√ n

Learning Objectives

Sampling Distributions
Of The Mean

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


Learning Objectives

Sampling Distribution of
the Mean

To understand population, sample,
and sampling distribution.

Basic Concepts
1. A normal distribution
2. Mean equal to the population mean.

3. Standard deviation
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.


Learning Objectives

Sampling Distribution
Distributionsof
Ofthe
TheMean
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.

x

1

x

<

< x + 1

x


Learning Objectives

Sampling Distribution of
the Proportion

To recognize problems involving
sampling means and proportions.

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

=

√ P (1-P)

n


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


Learning Objectives

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


Learning Objectives

Determining Sample Size

To recognize problems involving
sampling means and proportions.

Problems Involving Proportions

n

=

Z2 [P1-P)]
E2


Learning Objectives

Determining Sample Size

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.


Learning Objectives

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.
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)


Learning Objectives

Determining Sample Size

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

Learning Objectives

Determining Sample Size

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


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


Learning Objectives

The End

Copyright © 2004 John Wiley & Sons, Inc.