Chapter 13

Determining the Size of

a Sample

Sample Accuracy

• Sample accuracy: refers to how close a random sample’s statistic ( mean, S.D. ) is to the true population’s value ( µ, š ) it represents

• Important points:• Sample size is not related to

representativeness• Sample size is related to accuracy

Sample Size Axioms

• The only perfectly accurate sample is a census.• A probability sample will always have some

inaccuracy (sample error).• The larger a probability sample is, the more

accurate it is (less sample error).• Probability sample accuracy (error) can be

calculated with a simple formula, and expressed as a ± % number.

Sample Size Axioms…cont.

• You can take any finding in the survey, replicate the survey with the same probability sample size, and you will be “very likely” to find the same finding within the +- range of the original finding.

• In almost all cases, the accuracy (sample error) of a probability sample is independent of the size of the population.

Sample Size Axioms…cont.

• A probability sample can be a very tiny percentage of the population size and still be very accurate (have little sample error). (e.g., consumer confidence panel– 5,000 )

• The size of the probability sample depends on the client’s desired accuracy (acceptable sample error) balanced against the cost of data collection for that sample size.

The Confidence Interval Method of Determining Sample Size

• Confidence interval approach: applies the concepts of accuracy, variability, and confidence interval to create a “correct” sample size

• Two types of error:• Nonsampling error: pertains to all sources of error

other than sample selection method and sample size• Sampling error: involves sample selection and

sample size• Sample error formula:

The Confidence Interval Method of Determining Sample Size

• The relationship between sample size and sample error:

The Confidence Interval Method of Determining Sample Size

Variability

• Variability: refers to how similar or dissimilar responses are to a given question

• P: percent• Q: 100%-P

The Confidence Interval Method of Determining Sample Size

Normal Distribution

The Confidence Interval Method of Determining Sample SizeNotion of Confidence Interval

• Confidence interval: range whose endpoints define a certain percentage of the responses to a question

• Central limit theorem: a theory that underlies many statistical concepts and holds that a bar chart of all repeated samples of a survey within a population would look like a normal curve

Sample Size Formula

• To calculate the proper sample size for a survey, only three factors need to be considered:• The amount of variability believed to be in the

population• The desired accuracy• The level of confidence required in your

estimates of the population values

Sample Size Formula…cont.

• Standard sample size formula:

Examples

• At 95% confidence level: e = ± 10

1.962 (50x50) 102

At 99% confidence level: e = ± 5 2.582 (50x50) 52

n = = 96

n2

= = 665.6

Practical Considerations in Sample Size Determination

• How to estimate variability in the population• Expect the worst case.• Estimate the actual variability.

• How to determine the amount of desired sample size

• How to decide on the level of confidence desired• How to balance sample size with the cost of data

collection

Other Methods of Sample Size Determination

• Arbitrary “percentage rule of thumb” sample size:• Arbitrary sample size approaches rely on

erroneous rules of thumb.• Arbitrary sample sizes are simple and easy to

apply, but they are neither efficient nor economical.

Other Methods of Sample Size Determination…cont.

• Conventional sample size specification• Conventional approach follows some

“convention” or number believed somehow to be the right sample size.

• Using conventional sample size can result in a sample that may be too large or too small.

• Conventional sample sizes ignore the special circumstances of the survey at hand.

Other Methods of Sample Size Determination…cont.

• Statistical analysis requirements of sample size specification• Sometimes the researcher’s desire to use

particular statistical technique influences sample size.

• Cost basis of sample size specification• “All you can afford” method• Instead of the value of the information to be gained

from the survey being a primary consideration in the sample size, the sample size is determined by budget factors that usually ignore the value of the survey’s results to management.

Special Sample Size Determination Situations

Sampling from Small Populations

• Small population: sample exceeds 5% of total population size

• Finite multiplier: adjustment factor for sample size formula

• Appropriate use of the finite multiplier formula will reduce a calculated sample size and save money when performing research on small populations

Finite Multiplier

k1 = (N - n)/(N – 1)

Where:

N = Population size

n = sample size

Special Sample Size Determination Situations

Sample Size Using Nonprobability Sampling

• When using nonprobability sampling, sample size is unrelated to accuracy, so cost-benefit considerations must be used

Case 13.1 Peaceful Lake Subdivision

• Please read Case 13.1 in pp. 396-397.

• Analyze the case and answer questions 1, 2.

Case 13.1

( pq )

e2

=384

n1 = 384 x Finite Multiplier

= 384 x 0.60

= 230

n = z2