1. 2.2 Systematic, Stratified, and Cluster Sampling

2. Systematic sampling
To perform systematic sampling, we randomly select an individual out of the first k individuals and also select every kth individual after the first selected individual.

3. Example: Design a Plan to Perform Systematic Sampling
A grocery store owner wants to survey customers as they leave the store on a certain day. There are about customers each day. The owner would like to get feedback from at least 40 customers. From past experience, she knows that if 60 customers are approached, about 40 of them will participate in the survey. Design a plan to select a sample.

4. Solution
First we divide 470 by 60 which is 7.8. If we round up to 8 and approach every 8th person, the number of people approached would be only be about (58 people). But that would not meet the owner’s goal to approach 60 people. If we round 7.8 down to 7 and approach every 7th customer, the number of people approached would be about 67 people, which meets the owner’s goal. So, every 7th customer should be approached.

5. Solution
To determine the first individual, we randomly select a number between 1 and 7, inclusive. Using the seed 58 (the current temperature outside the author’s home when writing this paragraph) with a TI-84, we obtain the number 2. So, the pollster should approach the 2nd customer and every 7th customer after that:
2, = 9, = 16, = 23, = 30, …
So, the pollster should approach the 2nd customer, the 9th customer, the 16th customer, the 23rd customer, the 30th customer, and so on.

6. Stratified sampling
To perform stratified sampling, a population is divided into subgroups called strata (singular: stratum) and simple random sampling is performed on each stratum. In each stratum, the individuals share some characteristic. The ratios of the number of individuals selected from each stratum are equal to the corresponding ratios of the stratum sizes.

7. Cluster sampling
To perform cluster sampling, a population is divided into subgroups called clusters. Then simple random sampling is used to select some of the clusters. All of the individuals in those selected clusters form the sample.

8. Example: Designing a Plan to Perform Cluster Sampling
The president of a college would like professors to survey 1000 of the college’s 35,000 students about their awareness of various services available on campus. Although the president would like the survey to be administered during class time, she would prefer that it not disrupt all classes. Assuming that the smallest class size is 20 students, design a survey plan.

9. Solution
Simple random sampling, systematic sampling, and stratified sampling would disrupt a lot of classes. If each class is treated as a cluster, then cluster sampling would disrupt far fewer classes. The first step would be to create a frame of all classes. Then simple random sampling can be used to select the classes. To determine the number of classes that should be selected, we divide the desired sample size, 1000 students, by the smallest class size, 20 students: 1000 ÷ 20 = 50 classes. All the students in those 50 classes should be surveyed.

10. Solution
The sample size will likely turn out to be more than the desired size of 1000 students because the minimum class size was used to calculate how many classes should be selected. To use a larger class size to perform the calculation runs the risk of ending up with fewer students than the desired sample size.

11. Requirements and Benefits of Sampling Methods

12. Convenience sampling
To perform convenience sampling, we gather data that are easy to collect and do not bother with collecting them randomly.

13. Voluntary response sampling
To perform voluntary response sampling, we let individuals choose to be in the sample.

14. Example: Identifying Sampling Methods
Identify whether the sampling method is simple random, systematic, stratified, cluster, or convenience. Explain.
1. A researcher randomly selects 20 Taco Bell locations and surveys all the employees at those locations.
2. A news station hosts a call-in survey about whether physician-assisted death should be legalized in all states.
3. A researcher randomly selects an LED TV out of the first 200 LED TVs on an assembly line and also selects every 200th LED TV after that.

15. Example: Identifying Sampling Methods
Identify whether the sampling method is simple random, systematic, stratified, cluster, or convenience. Explain.
4. In a study at a community college, 30 instructors are randomly selected from fulltime instructors and 50 instructors are selected from part-time instructors.
5. The City Hall of Spring Hill, Kansas, creates a frame of its 5730 residents and randomly selects 60 residents.

16. Solution
1. A researcher randomly selects 20 Taco Bell locations and surveys all the employees at those locations.
Because all the employees at 20 clusters were surveyed, cluster sampling was performed.
2. A news station hosts a call-in survey about whether physician-assisted death should be legalized in all states.
A call-in survey would involve only those who choose to call in. So, the station performed voluntary response sampling, which is one type of convenience sampling. No type of convenience sample should ever be performed.

17. Solution
3. A researcher randomly selects an LED TV out of the first 200 LED TVs on an assembly line and also selects every 200th LED TV after that.
Because every 200th TV was selected, systematic sampling was performed.
4. In a study at a community college, 30 instructors are randomly selected from fulltime instructors and 50 instructors are selected from part-time instructors.
Because instructors were randomly selected from (two) strata, stratified sampling was performed.

18. Solution
5. The City Hall of Spring Hill, Kansas, creates a frame of its 5730 residents and randomly selects 60 residents.
Because the residents were randomly selected from a single frame, simple random sampling was performed.