Learning Objectives Copyright © 2002 South-Western/Thomson Learning Sample Size Determination...
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Learning ObjectivesLearning Objectives
Copyright © 2002 South-Western/Thomson Learning
Sample Size Determination
CHAPTER
thirteen
CHAPTER
thirteen
Learning ObjectivesLearning 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
5. To distinguish between point and interval estimates.
6. To recognize problems involving sampling means and proportions.
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Financial, Statistical, and Managerial Issues
As a general rule:
The larger the sample, the smaller the sampling error.
Larger samples cost more.
The financial and statistical issues in the determination of sample size.
Determining Sample Size for Probability Samples
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Budget Available
Financial constraints challenge the researcher.
The financial and statistical issues in the determination of sample size.
Methods for Determining Sample Size
Rules of Thumb
• Desired sampling error
• Past experience
• Similar studies
• A gut feeling
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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.
To discover the methods for determining sample size.
Methods for Determining Sample Size
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General Properties
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.
To gain an appreciation of a normal distribution.
The Normal Distribution
Learning ObjectivesLearning Objectives
6. The area between the mean and a given number of standard deviations from the mean is the same for all normal distributions.
The Standard Normal Distribution
• The same features as any normal distribution.
• The mean is equal to zero
• The standard deviation is equal to one.
To gain an appreciation of a normal distribution.
The Normal Distribution
Learning ObjectivesLearning Objectives
Z=value of the variable - mean of the variable
standard deviation of the variable
To gain an appreciation of a normal distribution.
Z =X -
where
X = value of the variable
= mean of the variable
= standard deviation of the variable
The Normal Distribution
Learning ObjectivesLearning Objectives
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
To understand population, sample, and sampling distributions.
Population, Sample, and Sampling Distributions
Learning ObjectivesLearning Objectives
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:
To understand population, sample, and sampling distributions.
s x =n√
Population, Sample, and Sampling Distributions
Learning ObjectivesLearning Objectives
The Standard Error of the Mean
Applies to the standard deviation of a distribution of sample means.
To understand population, sample, and sampling distribution.
x =n√
Population, Sample, and Sampling Distributions
Learning ObjectivesLearning Objectives
Basic Concepts
1. A normal distribution2. 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.
Sampling Distribution of the Mean
To understand population, sample, and sampling distribution.
Learning ObjectivesLearning Objectives
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.
To understand population, sample, and sampling distribution.
Sampling Distribution of the Mean
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Point Estimates Inferences regarding the sampling error associated with a particular estimate of the population value.
Interval EstimateInference regarding the likelihood that a population value will fall within a certain range.
To distinguish between point and interval estimates.
1 xx < < x + 1 x
Sampling Distribution of the Mean
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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:
To recognize problems involving sampling means and proportions.
Sampling Distribution of the Proportion
P (1-P)Sp =
n√
Learning ObjectivesLearning Objectives
where:
Sp = standard error of sampling distribution
proportion
P = estimate of population proportion
n = sample size
To recognize problems involving sampling means and proportions.
Sampling Distribution of the Proportion
P (1-P)Sp =
n√
Learning ObjectivesLearning Objectives
Problems Involving Means
The formula for calculating the required sample size for problems that involve the estimation of a mean:
Sample Size Determination
n =Z2 2
E2
where:Z = level of confidence expressed in
standard errors
= population standard deviation
E = acceptable amount of sampling error
To recognize problems involving sampling means and proportions.
Learning ObjectivesLearning Objectives
Problems Involving Proportions
n =Z2 [P1-P)]
E2
To recognize problems involving sampling means and proportions.
Sample Size Determination
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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)
To recognize problems involving sampling means and proportions.
Sample Size Determination
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Adjusting for a sample that is 5 percent or more of the population and dropping the independence assumption:
x =n√
N - n√ N - 1
To recognize problems involving sampling means and proportions.
Sample Size Determination
Learning ObjectivesLearning Objectives
Reducing the required sample size using the Finite Population Correction
nN
N + n -1n' =
where:n' = revised sample size
n = original sample size
N = population size
To recognize problems involving sampling means and proportions.
Sample Size Determination
Learning ObjectivesLearning Objectives
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.
To recognize problems involving sampling means and proportions.
Sample Size Determination
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Statistical PowerThe 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.
Statistical Power To recognize problems involving sampling means and proportions.
Learning ObjectivesLearning Objectives
• 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
SUMMARY
Learning ObjectivesLearning Objectives
The End
Copyright © 2002 South-Western/Thomson Learning
