Chapter Thirteen Copyright © 2004 John Wiley & Sons, Inc. Sample Size Determination

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Copyright © 2004John Wiley & Sons, Inc.

Sample Size Determination

John Wiley & Son, Inc 2

Learning Objective

1. To learn the financial and statistical issues in the determination of sample size

2. To discover methods for determining sample size

3. To gain an appreciation of a normal distribution

Learning Objectives


Learning Objective

4. To understand population, sample, and sampling distributions.

5. To distinguish between point and interval estimates

6. To recognize problems involving sampling means and proportions.

Learning Objectives


Learning ObjectiveThe financial and statistical issues in the

determination of sample size.

Determining Sample Size for Probability Samples

• Financial, Statistical, and Managerial Issues– As a general rule:

– The larger the sample, the smaller the sampling error.

– Larger samples cost more on a liner basis; 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.


Learning ObjectiveDetermining Sample Size for Probability Samples The financial and statistical issues in the


• 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 ObjectiveDetermining Sample Size for Probability Samples The financial and statistical issues in the


• 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 ObjectiveThe Normal DistributionTo 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



• Important Characteristics of the Normal Distribution1. The normal distribution is bell-shaped and has only one

mode.2. Symmetrical about the mean3. 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



• 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 Objective

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

The Normal DistributionTo gain an appreciation of a normal distribution.


Learning ObjectiveTo understand population, sample,

and sampling distributions.

Population and Sample 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 Objective

s x =n√

Sampling Distributions Of The Mean To understand population, sample,


• 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:


Learning ObjectiveSampling Distributions Of The Mean To understand population, sample,


• Basic Concepts– Characteristics of a Large Simple Random

Sample• 30 or more observations• Distribution is a normal distribution• Distribution has a mean equal to the population

mean• Distribution has a standard deviation, referred to as

the standard error of the mean, equal to the population standard deviation divided by the square root of the sample


Learning Objective

• The Standard Error of the Mean– Applies to the standard deviation of a distribution

of sample means

x =n√

To understand population, sample, and sampling distributions.

Sampling Distributions Of The Mean


Learning ObjectiveSampling Distribution of the Mean To understand population, sample,


• Making Inferences on the Basis of a Single Sample– A 68.26% probability that a random sample will produce

an estimate of the population mean within one standard error (±) of the true mean

– 95.44 % probability that a random sample will produce an estimate of the population mean within two standard error (±) of the true mean

– 99.74 % probability that a random sample will produce an estimate of the population mean within three standard error (±) of the true mean


Learning ObjectiveTo distinguish between point

and interval estimates.


• Point Estimates – Inferences regarding the sampling error associated with a

particular estimate of the population value.• Sampling Error

– The distance between the x and Χ• Interval Estimate

– The likelihood that a population value falls within a certain range.

• Confidence Interval– The interval that in all probability, includes the true

population value


Learning Objective

• Method of Deriving Interval Estimates– Draw a random sample from the population

interest and calculating the mean of that sample– There is a 68.26% probability that the sample

mean lies within one σ x

– Below is the statement symbolically

1 xx < < x + 1 x

To distinguish between point and interval estimates.



Learning ObjectiveTo recognize problems involving sampling

means and proportions.

Sampling Distribution of the Proportion

• Interest in Estimating Proportions or Percentages– Rather than or in addition to estimating means.

Examples• The percentage of the population that is aware of a

particular

• The percentage of the population that accesses the Internet one or more times in an average week

• The percentage of the population that has visited a fast-food restaurant four or more times in the past 30 days


Learning ObjectiveTo recognize problems involving sampling

means and proportions.

Sampling Distribution of the Proportion

P (1-P)Sp =

n√

• 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:


Learning Objective

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

To recognize problems involving sampling means and proportions.

Determining Sample Size


Learning ObjectiveDetermining Sample SizeTo recognize problems involving sampling means and proportions.

• Four Methods Used to Deal with the Estimate of the Population Standard Deviation

1. Use results from a prior survey

2. Conduct a pilot survey

3. Use secondary date

4. Use judgment



• Problems Involving Proportions– If there is no basis for estimating P, the researcher

can make the most pessimistic case regarding the value of P

– Given the values of Z and E, what value of P will require the largest possible sample

– A value of .50 will make the value of the expression P(1-P) larger than any possible value of P.



• 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.

x =n√

N - n√ N - 1



– 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)

nN

N + n -1n' =

where:n' = revised sample size

n = original sample size

N = population size


Learning Objective

• Types of Errors– Type I Error

• The error of concluding that there is a difference when there is no difference

– The formulas in this chapter focus on this type of error

– Type II Error• The error of saying that there is no difference when

there is a difference

– Statistical Power• The probability of not making a Type II error

Statistical Power To recognize problems involving sampling means and proportions.


Learning Objective

• Determining Sample Size for Probability Samples

• The Normal Distribution

• Population and Sample Distributions

• Sampling distribution of the Mean

• Determining Sample Size

• Statistical Power

SUMMARY


Learning Objective

The End

Copyright © 2004 John Wiley & Sons, Inc.


Learning Objective

• 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 Objective

The End

Copyright © 2004 John Wiley & Sons, Inc.

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Chapter Thirteen Copyright © 2004 John Wiley & Sons, Inc. Sample Size Determination