1.6 More On Sampling Design

Sometimes a strict simple random sample is difficult to obtain; therefore, we need to find other ways of randomly selecting units for a sample.

Multistage Sampling Design:

This type of sampling is used to select a sample from a very large population where certain groups and subgroups are available.

Example

Suppose we wish to obtain a sample of people in the United States.

1. Randomly select a few states (SRS)
2. Randomly select a few counties (SRS)
3. Randomly select a few neighborhoods (SRS)
4. Randomly select people (SRS)

At each stage, we obtained a sampling frame (list of states; list of counties in the selected states; list of neighborhoods in the selected counties; list of people in the selected neighborhoods) and selected a simple random sample from that sampling frame. This method (multistage sampling) is much easier that selecting our sample from a list of all Americans (if such a list could be found).

Systematic Random Sampling Design:

This method is performed by randomly selecting a starting point in the sampling frame and then selecting every n^th unit to be in the sample until the desired sample size has been reached. Usually every 5^th or every 10^th unit is selected.

Stratified Random Sampling Design:

This method divides the sampling frame into groups that are of interest.

1. Divide the sampling frame into groups of units called strata. The strata are chosen using some characteristic of the units that is already known (i.e.: race, gender, location, etc.). The strata are chosen because we have a special interest in these groups within the population or because the units in each stratum resemble each other.
   (Note: strata is plural, while stratum is singular.)
2. Take a separate simple random sample in each stratum and combine these to make up the stratified random sample.

Note: The major way that stratified samples differ from simple random samples is that stratified samples need not give all units in the population the same chance of being chosen.

Example

A university has 30,000 students of whom 3,000 (10%) are black. Using an SRS of 500 students, we would expect 50 (10%) of the students in the sample to be black. This size has low precision (the opinions of the black students will be underrepresented), so we must increase the number of black students in the sample to increase this precision. So instead of the SRS, we'll randomly select 200 black students and 300 other students by dividing the registrar's list of students into two lists (one of black students and one of other students), and then we'll take an SRS of 200 from the list of black students and an SRS of 300 from the list of other students.

• The chance of any one black student being chosen is:
  

• The chance of any other student being chosen is:
  

It is true that the blacks seem to be over-represented in our sample, but since the probability of selection for each group is known, we can correct for the over-representation when we analyze our data; when we know the probabilities associated with each stratum, we have a probability sample.

Definition

Probability Sample

A probability sample is a sample chosen in such a way that we know what samples are possible and what chance, or probability, each possible sample has to be chosen (not all need be equally probable).

Example

Suppose we sample 200 black students and 300 other students by the method of the preceding example and ask each of the students, "Do you favor the creation of a new degree program in African studies?" In the sample, 162 of the black students and 174 of the other students say that they are interested in the new African studies degree program. We then calculate, for each stratum, the statistic that will estimate the proportion of the students that are interested in the new program.

We estimate that or 81% of the black students favor the new program, while about or 58% of the other students favor the new program. To estimate the proportion of all students at the university who favor the program, first estimate how many students favor the new program as follows:

In all, we estimate that 2,430 + 15,660 = 18,090 of the 30,000 students at the university are in favor of the new degree program. That is, of all students at the university, about or 60% favor the new degree program. The stratified sample allows us to say that about 81% of the black students, 58% of the other students, and 60% of all students favor the creation of an African studies degree program.

As you can see, stratified samples (and likewise, probability samples) give us more information than we can get from just a simple random sample.

Homework

(Scott Street's section only)

Pages 51-54

1.52, 1.58 (a-c)

(Solutions)

Please direct all questions regarding STAT-110 to your instructor or to the director of STAT-110, Dr. Todd Ogden at ogden@stat.sc.edu.

Mail comments regarding this presentation to W. Scott Street, IV at street@stat.sc.edu.

© 1996 by W. Scott Street, IV