1. Determining the Size of
Chapter 13
Determining the Size of
a Sample

2. 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

3. 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.

4. 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.

5. 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.

6. 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:

7. The Confidence Interval Method of Determining Sample Size
The relationship between sample size and sample error:

8. 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

9. The Confidence Interval Method of Determining Sample Size
Normal Distribution

10. The Confidence Interval Method of Determining Sample Size
Notion 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

11. 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

12. Sample Size Formula…cont.
Standard sample size formula:

13. Examples At 95% confidence level: e = ± 10 1.962 (50x50) 102 n = = 9652
n = = 96 n2
= = 665.6

14. 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

15. 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.

16. 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.

17. 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.

18. 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

19. Finite Multiplier k1 = (N - n)/(N – 1) Where: N = Population size
n = sample size

20. 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

21. Case 13.1 Peaceful Lake Subdivision
Please read Case 13.1 in pp
Analyze the case and answer questions 1, 2.

22. Case 13.1 ( pq ) e2 =384 n1 = 384 x Finite Multiplier = 384 x 0.60
= 230
n = z2