Chapter 11 Sampling Design Chapter Objectives define sampling, sample, population, element, subject...

Chapter 11Sampling Design

Chapter Objectives

• define sampling, sample, population, element, subject and sampling frame

• describe and discuss the different probability and non-probability sampling designs

• identify the use of appropriate sampling designs for different research purposes

• discuss precision and confidence • estimate sample size • discuss efficiency in sampling • discuss generalisability in the context of sampling

designs

The Principles of Sampling Design

Population, Element, Sampling Frame, Sample and Subject

• Population (or target population)• entire group of people, events or things of interest that

the researcher wishes to investigate

• Element• a single member of the population

• Sampling Frame• a listing of all the elements in the population from which

the sample is drawn

• Sample• a subset of the population

• Subject• a single member of the sample

Relationship between Population, Sampling Frame

and Sample

Relationship between Sample Statistics

and Population Parameters

Advantages of Sampling

• Less costs– cheaper than studying whole population

• Less errors due to less fatigue– better results

• Less time– quicker

• Destruction of elements avoided– eg bulbs

Normal Distibution in a Population

As the sample size n increases, the means of the random samples taken from practically any population approach a normal distribution with mean μ and standard deviation

Representativeness of Samples• If the sample mean is much > than the

population mean μ then the sample would overestimate the true population mean

• If the sample mean is much < than the population mean μ then the sample would underestimate the true population mean

• The more representative the sample is of the population, the more generalisable are the findings of the research.

Preparing a Sampling Design

Probability & Non-probability Sampling

• Probability Sampling– the elements in the population have some known

chance or probability of being selected as sample subjects

• Non-probability Sampling– the elements do not have a known or

predetermined chance of being selected as subjects

Probability Sampling

• Simple random sampling– every element in the population has a known and

equal chance of being selected as a subject

• Complex (or restricted) probability sampling– procedures to ensure practical viable alternatives

to simple random sampling, at lower costs, and greater statistical efficiency

Simple Random Sampling

• Is the most representative of the population for most purposes

• Disadvantages are:– Most cumbersome and tedious– The entire listing of elements in population

frequently unavailable– Very expensive– Not the most efficient design

Complex Probability Sampling

• Systematic sampling

• Stratified random sampling

• Cluster sampling

• Area sampling

• Double sampling

Systematic Sampling• Every nth element in the population starting

with a randomly chosen element

• Example:– Want a sample of 35 households from a total of 260

houses. Could sample every 7th house starting from a randomly chosen number from 1 to 10. If that random number is 7, sample 35 houses starting with 7th house (14th house, 21st house, etc)

– Possible problem is that there could be systematic bias. eg every 7th house could be a corner house, with different characteristics of both house and dwellers.

Stratified Random Sampling• Comprises sampling from populations segregated

into a number of mutually exclusive sub-populations or strata. Eg– University students divided into juniors, seniors, etc– Employees stratified into clerks, supervisors, managers,

etc

• Homogeneity within stratum and heterogeneity between strata

• Statistical efficiency greater in stratified samples• Sub-groups can be analysed• Different methods of analysis can be used for

different sub-groups.

Stratified Random Sampling Example

Stratum Motivation LevelClerks LowMiddle ManagersVery highTop Managers Medium

Combined X would not discrimate among groups

• Stratified Sampling– Proportionate sampling– Disproportionate sampling

Proportionate & Disproportionate

Stratified Random Sampling

Cluster Sampling• Take clusters or chunks of elements for study

– Eg, sample all students in MGMT 303 and MGMT 304 to study the characteristics of Management Science majors

• Advantage of cluster sampling is lower costs• Statistically it is less efficient than other

probability sampling procedures discussed so far

Area Sampling:• Cluster sampling confined to a particular area

– Eg, sampling residents of a particular locality, county, etc

Double Sampling

• Collect preliminary data from a sample, and choose a sub-sample of that sample for more detailed investigation.

• Example:– Conduct unstructured interviews with a

sample of 50. – Repeat a structured interview with 30 from

the 50 originally sampled.

Non-probability Sampling

• Convenience sampling– Survey whoever is easily available– Used for quick diagnosis of situations

• Simplest and cheapest• Least reliable

• Purposive sampling– Judgement sampling– Snowball sampling– Quota sampling

Judgement Sampling

• Involves the choice of subjects who are in the best position to provide the information required

• Experts’ opinions could be sought– Eg, Doctors surveyed for cancer causes

Snowball Sampling

• Used when elements in population have specific characteristics or knowledge, but are very difficult to locate and contact.

• Initial sample group can be selected by probability or non-probability methods, but new subjects are selected based on information provided by initial subjects. – Eg, used to locate members of different

stakeholder groups regarding their opinions of a new public works project.

Quota Sampling

• Quotas for numbers or proportion of people to be sampled, established.

• Examples:1) survey for research on dual career

families: 50% working men and 50% working women surveyed.

2) Women in management survey: 70% women surveyed and 30% men surveyed.

Choice Points in Sampling Design

Precision and Confidence• Precision

– refers to how close the sample estimate eg X is to the true population characteristic( ) depends on the variablity in the sampling distribution of the mean, ie the standard error ( S X )

– indicates the confidence interval within which the population mean can be estimated (= X + KS X )

• Confidence– reflects the level of certainty that the sample

estimates will actually hold true for the population– bias is absent from the data– accuracy is reflected by the confidence level ( K )

XS

Standard Error

S Sn

X

S X

S = standard deviation of the samplen

= sample size

= standard error or standard deviation of the sample mean

Characteristics of the Standard Error

• The smaller the standard deviation of the population, the smaller the standard error and the greater the precision

• The standard error varies inversely with the square root of the sample size. Hence the larger the n, the smaller the standard error, and the greater the precision.

S Sn

X

Confidence Interval for the Mean

X KS XX

S X

K

= population mean

= sample mean

= standard error

= z statistic for large samples ≥ 30

= t statistic for small samples < 30

Confidence Levels

• For large samples, K = z score= 1.65 for 90% confidence level= 1.96 for 95% confidence level= 2.58 for 99% confidence level

• Example: a 95% confidence interval for mean purchases (μ) by customers based on a sample mean of $105 with a standard error of $1.43 is:

μ = 105 ± 1.96*1.43 = 105 ± 2.80 Hence μ would fall between $102.20 and

$107.80

X KS X

Trade-off between Precision and Confidence

Determining the Sample Size

X KS X

Example: Suppose a manager wants to be 95% confident that withdrawals from a bank will be within a confidence level of ± $500. From a sample of customers the standard deviation S was calculated as $3500. What sample size is needed?

The expression is equivalent to the precision or admissible margin of error. Let this be E.

KS X

E KS Xor E K S

n *

Determining the Sample Size (cont’d)

Rearranging these terms, a formula for the sample size n is:

nK S

E

* 2

Substituting K=1.96 (95% confidence), S=3500, and E=500 into this equation, provides the sample size n:

n

n

n

1 3500

500

2

13

188

2

.96*

.72

Roscoe’s Rules of Thumb for Determining Sample Size

• Sample sizes larger than 30 and smaller than 500 are appropriate for most research

• Minimum sample size of 30 for each sub-category is usually necessary

• In multivariate research, the sample size should be several times as large as the number of variables in the study

• For simple experimental research, successful research is possible with samples as small as 10 to 20

Efficiency in Sampling

If n is constant, you should get a smaller

or

For the same , you should use a

smaller n

S X

S X

Review of Sample Size Decisions

• How much precision is wanted in estimating the population characteristics, ie what is the margin of admissible error or confidence interval?

• How much confidence is really needed. How much risk can we take of making errors in estimating the population parameters (ie confidence level)?

• How much variability is in the population? The greater the variability, the larger the sample size needed.

• Cost and time constraints• The size of the population (N) itself