sample size

Learning ObjectivesLearning Objectives

Copyright © 2004John Wiley & Sons, Inc.

Sample Size Determination

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CHAPTER

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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

5. To distinguish between point and interval estimates.

6. To recognize problems involving sampling means and proportions.


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.

The financial and statistical issues in the determination of sample size.

Determining Sample Size for Probability Samples


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

The financial and statistical issues in the determination of sample size.

Methods for Determining Sample Size

Rules of Thumb

- Desired sampling error - Similar Studies

- Past experience - A gut feeling


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


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

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

To gain an appreciation of a normal distribution.

The 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.




Z=value of the variable - mean of the variable

standard deviation of the variable


Z =X -

where

X = value of the variable

= mean of the variable

= standard deviation of the variable



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.

Sampling Distributions Of The Mean


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√



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√



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.


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



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√


where:

Sp = standard error of sampling distribution

proportion

P = estimate of population proportion

n = sample size


Sampling Distribution of the Proportion

P (1-P)Sp =

n√


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


Determining Sample Size


Problems Involving Proportions

n =Z2 [P1-P)]

E2




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.




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)




Adjusting for a sample that is 5 percent or more of the population and dropping the independence assumption:

x =n√

N - n√ N - 1




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




• 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

sample size

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