Sample Size
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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
Methods for Determining Sample Size
Learning Objectives 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.
The Normal Distribution
Learning Objectives
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 bellshaped 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
The Normal Distribution
Z=
To gain an appreciation of a normal distribution.
value of the variable mean of the variable
where
standard deviation of the variable
X µ Z = σ
X = value of the variable µ = mean of the variable σ = standard deviation of the variable
Sampling Distributions Of The Mean
Learning Objectives 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
Sampling Distributions Of The Mean
Learning Objectives 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 µ and a standard deviation equal to:
s x
=
σ
√ 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.
σ x
=
σ
√ n
Sampling Distribution of the Mean
Learning Objectives 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.
Sampling Distribution of Sampling Distributions Of The Mean the Mean
Learning Objectives 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
1σ x < µ < x + 1σ 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 (1P) n
Learning Objectives
Sampling Distribution of the Proportion
Sp
=
To recognize problems involving sampling means and proportions.
√ P (1P) 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 σ2 E2
where: Z = level of confidence expressed in standard errors
σ = population standard deviation E = acceptable amount of sampling error
Determining Sample Size
Learning Objectives To recognize problems involving sampling means and proportions.
Problems Involving Proportions
n
=
Z2 [P1P)] E2
Determining Sample Size
Learning Objectives 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.
Determining Sample Size
Learning Objectives 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. (Nn) / (N1)
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:
σ x
=
σ
√ n
√
N n N 1
Determining Sample Size
Learning Objectives 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
SUMMARY
Learning 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
Learning Objectives
The End
Copyright © 2004 John Wiley & Sons, Inc.
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
Methods for Determining Sample Size
Learning Objectives 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.
The Normal Distribution
Learning Objectives
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 bellshaped 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
The Normal Distribution
Z=
To gain an appreciation of a normal distribution.
value of the variable mean of the variable
where
standard deviation of the variable
X µ Z = σ
X = value of the variable µ = mean of the variable σ = standard deviation of the variable
Sampling Distributions Of The Mean
Learning Objectives 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
Sampling Distributions Of The Mean
Learning Objectives 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 µ and a standard deviation equal to:
s x
=
σ
√ 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.
σ x
=
σ
√ n
Sampling Distribution of the Mean
Learning Objectives 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.
Sampling Distribution of Sampling Distributions Of The Mean the Mean
Learning Objectives 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
1σ x < µ < x + 1σ 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 (1P) n
Learning Objectives
Sampling Distribution of the Proportion
Sp
=
To recognize problems involving sampling means and proportions.
√ P (1P) 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 σ2 E2
where: Z = level of confidence expressed in standard errors
σ = population standard deviation E = acceptable amount of sampling error
Determining Sample Size
Learning Objectives To recognize problems involving sampling means and proportions.
Problems Involving Proportions
n
=
Z2 [P1P)] E2
Determining Sample Size
Learning Objectives 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.
Determining Sample Size
Learning Objectives 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. (Nn) / (N1)
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:
σ x
=
σ
√ n
√
N n N 1
Determining Sample Size
Learning Objectives 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
SUMMARY
Learning 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
Learning Objectives
The End
Copyright © 2004 John Wiley & Sons, Inc.
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