Population and Sampling

1
Population and Sampling

• AED 616
• Spring 2007

2
What is a Population?

• In Social Science research, the investigator
  seeks to obtain data from a specific source.
• In survey research, the target group is referred
  to as the population.
• The population has certain characteristics of
  interest to the researcher.
• When we gather data from an entire population, it
  is referred to as a census.

3
Limiting Factors

• What happens when a population is very large?
• Example of a population all students enrolled
  at the University of Arizona
• Another example of a population all high school
  students enrolled in agricultural education
  programs in Arizona
• A manageable population might be all
  agricultural education teachers in Arizona.

4
Limiting Factors (continued)

• When dealing with populations, the statistical
  representation is the upper case letter N.
• What happens when the population is too large for
  the researcher to deal with?
• Two limiting factors of conducting research are
• Time
• Funding
• A M.S. student may not have the time to complete
  a research project with a population of 500
  1,000 people.

5
Solution?

• When the population is too large, the researcher
  can sample the population.
• A sample is a subset of the population to which
  the researcher wants the sample, or the
  individuals actually involved in the research, to
  be representative of the population.
• Selecting a random sample ensures
  representativeness from a mathematical
  perspective.
• Sometimes, it is not possible to select random
  samples for all possible education research
  studies so, at times, purposeful samples are
  used.
• A sample either is or is not random.
• Random sampling must incorporate some aspect of
  random selection.

6
Random Sampling

• A random sample involves what is called
  probability sampling.
• Every member of the population has a nonzero
  probability of being selected for the sample.
• A random sample is an unbiased sample, which
  means that those individuals selected vary only
  as they would due to random fluctuation.

7
Random Selection vs Random Assignment

• A researcher at the UA selects a random sample
  of 250 from the freshman class of 6,821 students
  who are then surveyed about their attitudes
  toward certain factors of college life.
• This example involves random selection. The 250
  students of the sample are representative of the
  6,821 in the freshman class.

8

• An Ag Ed researcher is conducting an experiment
  on the use of hearing protection devices by
  agricultural educators in shops and labs.
• 80 teachers are participating in the research and
  20 will be randomly assigned to each type of
  hearing protection group.
• In this way, the four groups of teachers assigned
  to the different hearing protection device vary
  only on a random basis.
• As teachers are assigned, any teacher has the
  same probability of being assigned to any one of
  the four hearing protection device groups.
• Mainly, one in four, or one-quarter.

9
Use of a Random Number Table

• A simple random sample can be obtained by using a
  table of random numbers.
• Each member of a finite population is assigned a
  number, and then as many numbers as comprise the
  sample size are selected from the table.
• If there is a population of 70 members and 10 are
  to be selected at random, each of the 70 members
  is assigned a number from 1 to 70.
• The first 10 numbers that appear, wherever one
  begins in the random number table, determine the
  10 sample members.
• As there are only 70 members in the population,
  two-digit random numbers are used.
• Looking at an example table
• Beginning with the first row and going across,
  taking two-digit numbers in sequence gives the
  following 10 numbers
• 59, 39, 15, 80 (ignored, too high a number), 30,
  52, 09, 88 (ignored), 27, 18, 87 (ignored), 02,
  and 48.
• A number that is greater than your highest number
  (i.e.70) is ignored, as well as numbers that are
  repeated because a single member of the
  population is not included twice in the sample.
• Any kind of sequencing in the table is random.
• Not necessary to go across rows.
• Numbers could be selected in columns or by
  blocks.
• Random number generators are available on-line.

10
Sampling Error and Sampling Bias

• Sampling error is associated with random sampling
  and the term error does not mean making a
  mistake.
• Suppose we have a population of 1,675 fifth
  graders and we select a random sample of 150 from
  this population.
• The sample is administered a science achievement
  test and mean score on the test is 86.3
• Could it be said that the mean of the population
  is 86.3?
• No, but we are confident that the mean is a value
  around 86.3
• The difference between the sample mean of 86.3
  and the population mean (what ever it is) is an
  example of sampling error.
• Sampling error is related to variation.
• Sampling error is variation due o random
  fluctuation.
• As sample size increases, the variation due to
  random fluctuation and sampling error decreases.

11

• Bias enters in when a sample fails to represent
  the population is was intended to represent.
• Bias can be due to any number of sources, and it
  is a threat whenever nonrandom (or non
  probability) sampling is used, or when random
  sampling is used with a bias source.
• Referring to the fifth-grade population once more.

12

• Suppose instead of selecting a random sample, we
  selected five classes of fifth-graders, each
  class form a different school.
• Class average 30 students per class, so again
  have 105 students in the sample.
• But the classes selected are high-ability classes
  within the schools
• For this sample the mean on the science
  achievement test is 103.8.
• Does this mean represent the science achievement
  of the population?
• Sampling bias has occurred and this is a biased
  sample because of the way in which the sample was
  selected, in this case a nonrandom manner.

13

• Though random sampling was used, sampling bias
  can occur if we have a biased source.
• A most commonly cited survey in which a biased
  source was used was the 1936 survey by the
  Literary Digest.
• Predicted that Alf Landon would win the
  presidential election over Franklin Roosevelt
• Random sampling was used but the sample selected
  primarily from telephone directories and
  automobile registrations lists.
• In 1936, these lists were not representative of
  the voting population.
• Suppose you wanted to select a sample of the
  general population of U.S. adults.
• Country club memberships lists are used for the
  source and a random sample is selected from the
  list.
• Any problems?

14
Criteria for Sampling Design

• A good sampling design should meet four criteria
• Goal orientation. The sampling design should be
  tailored to the research design and be based on
  the studys goals and objectives.
• Measurability. Sampling design provides the data
  for the necessary analyses.
• Practicality. The actual activities of applying
  the sampling design have been identified and are
  feasible in the real situation.
• Economy. Research objectives can be met with
  available resources time, financial, personnel.

15
Stratified Random Sampling

• In some cases, a population may contain several
  sub populations.
• The researcher might divide the population into
  two or more subpopulations called strata.
• All strata are represented in the sample, and the
  sample members are selected from each stratum at
  random.
• Equal allocation
• Proportional allocation

16
Example

• Director of institutional research is conducting
  a survey of student opinion on the adequacy of
  facilities.
• Selects a 5 stratified random sample.
• University contains seven colleges, with a total
  enrollment of 15, 823 students.
• College is the stratifying variable, proportional
  allocation is used.
• 5 sample selected, sampling fraction is 1/20, .05

17
Sample Selection Using Proportional Allocation
University Example
18
What are other examples of strata that you can
think of?
19
Cluster Sampling

• Involves the random selection of clusters from
  the larger population of clusters.
• All the population members of a selected cluster
  are included in the sample.
• Classrooms are examples of cluster sampling.
• Clusters from the sample are randomly selected
  from the larger population of clusters.

20
Cluster Example

• Survey of fourth-grade achievement in
  mathematics, using a standardized achievement
  test.
• School system contains 33 elementary schools.
• Too expensive to administer test to all
  fourth-graders in the system.
• Classes are naturally assembled.
• All students in a class are selected
• If there are 83 fourth grade classes in the
  system, an average enrollment of 27 students.
• A sample size of 550 is desired.
• Decided to select 20 classes or clusters.