Question

Mrs. Inwood teaches a 7^th grade mathematics class who is interested in how many hours of homework a typical student from their middle school does in a week. The middle school contains grades 6-8 and has 275 students in total. Her students brainstormed plans on how to select a sample of students to survey. Listed below are several of the plans.

Provide a thorough analysis of the appropriateness (advantages and/or disadvantages) of each suggested sampling plan.

1.     Survey every fourth student (Student #4, Student #8, etc.) on each homeroom class list.

2.     Since there are 275 students in the middle school, put 250 white chips and 25 blue chips in a box. As students enter the cafeteria for lunch, have each one take a chip from the box. Survey the 25 students who get a blue chip.

3.     Have each student in Mrs. Inwood’s class conduct the survey in his or her history class.

4.     Distribute surveys to students as they enter the middle school building in the morning and ask students to return their completed surveys to the office at the end of the day.

Answer 1

1. This is a type of systematic sampling. In systematic sampling, every Nth name is selected from the list of the members of the target population. For instance, the sample will include the participants listed in every 10th from the list. That means the 10th, 20th, 30th and so on will be selected to become the members of the sample group.

In systematic random sampling, the researcher first randomly picks the first item or subject from the population. Then, the researcher will select each n'th subject from the list.

The procedure involved in systematic random sampling is very easy and can be done manually. The results are representative of the population unless certain characteristics of the population are repeated for every n'th individual, which is highly unlikely.

The process of obtaining the systematic sample is much like an arithmetic progression.

1. Starting number:
   The researcher selects an integer that must be less than the total number of individuals in the population. This integer will correspond to the first subject.
2. Interval:
   The researcher picks another integer which will serve as the constant difference between any two consecutive numbers in the progression.

   The integer is typically selected so that the researcher obtains the correct sample size

For example, the researcher has a population total of 100 individuals and need 12 subjects. He first picks his starting number, 5.

Then the researcher picks his interval, 8. The members of his sample will be individuals 5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93.

Other researchers use a modified systematic random sampling technique wherein they first identify the needed sample size. Then, they divide the total number of the population with the sample size to obtain the sampling fraction. The sampling fraction is then used as the constant difference between subjects.

Advantages of Systematic Sampling

• The main advantage of using systematic sampling over simple random sampling of system or process into the random selection of subjects.
• Another advantage of systematic random sampling over simple random sampling is the assurance that the population will be evenly sampled. There exists a chance in simple random sampling that allows a clustered selection of subjects. This is systematically eliminated in systematic sampling.

Disadvantage of Systematic Sampling

• The process of selection can interact with a hidden periodic trait within the population. If the sampling technique coincides with the periodicity of the trait, the sampling technique will no longer be random and representativeness of the sample is compromised.

2. This a type of Random Sampling. In this technique, each member of the population has an equal chance of being selected as subject. The entire process of sampling is done in a single step with each subject selected independently of the other members of the population.

There are many methods to proceed with simple random sampling. The most primitive and mechanical would be the lottery method. Each member of the population is assigned a unique number. Each number is placed in a bowl or a hat and mixed thoroughly. The blind-folded researcher then picks numbered tags from the hat. All the individuals bearing the numbers picked by the researcher are the subjects for the study. Another way would be to let a computer do a random selection from your population. For populations with a small number of members, it is advisable to use the first method but if the population has many members, a computer-aided random selection is preferred.

Advantages of Simple Random Sampling

One of the best things about simple random sampling is the ease of assembling the sample. It is also considered as a fair way of selecting a sample from a given population since every member is given equal opportunities of being selected.

Another key feature of simple random sampling is its representativeness of the population. Theoretically, the only thing that can compromise its representativeness is luck. If the sample is not representative of the population, the random variation is called sampling error.

An unbiased random selection and a representative sample is important in drawing conclusionsfrom the results of a study. Remember that one of the goals of research is to be able to make conclusions pertaining to the population from the results obtained from a sample. Due to the representativeness of a sample obtained by simple random sampling, it is reasonable to make generalizations from the results of the sample back to the population.

Disadvantages of Simple Random Sampling

One of the most obvious limitations of simple random sampling method is its need of a complete list of all the members of the population. Please keep in mind that the list of the population must be complete and up-to-date. This list is usually not available for large populations. In cases as such, it is wiser to use other sampling techniques.

3. This is an example of Stratified Sampling. Stratified sampling involves the use of “stratum”, or a subset of the target population wherein the members possess one or more common attribute. Examples of stratum include mothers, fathers, students, teachers, females, males, etc. Sampling error is usually lower in stratified sampling than in random sampling. Here, the students are in Mrs. Inwood's class.

Stratified sampling is a probability sampling technique wherein the researcher divides the entire population into different subgroups or strata, then randomly selects the final subjects proportionally from the different strata. Stratified sampling is a probability sampling technique wherein the researcher divides the entire population into different subgroups or strata, then randomly selects the final subjects proportionally from the different strata.

Uses of Stratified Random Sampling

• Stratified random sampling is used when the researcher wants to highlight a specific subgroup within the population. This technique is useful in such researches because it ensures the presence of the key subgroup within the sample.
• Researchers also employ stratified random sampling when they want to observe existing relationships between two or more subgroups. With a simple random sampling technique, the researcher is not sure whether the subgroups that he wants to observe are represented equally or proportionately within the sample.
• With stratified sampling, the researcher can representatively sample even the smallest and most inaccessible subgroups in the population. This allows the researcher to sample the rare extremes of the given population.
• With this technique, you have a higher statistical precision compared to simple random sampling. This is because the variability within the subgroups is lower compared to the variations when dealing with the entire population.

  Because this technique has high statistical precision, it also means that it requires a small sample size which can save a lot of time, money and effort of the researchers.

Types of Stratified Sampling

Proportionate Stratified Random Sampling

The sample size of each stratum in this technique is proportionate to the population size of the stratum when viewed against the entire population. This means that the each stratum has the same sampling fraction.

For example, you have 3 strata with 100, 200 and 300 population sizes respectively. And the researcher chose a sampling fraction of ½. Then, the researcher must randomly sample 50, 100 and 150 subjects from each stratum respectively.

Stratum	A	B	C
Population Size	100	200	300
Sampling Fraction	½	½	½
Final Sample Size	50	100	150

The important thing to remember in this technique is to use the same sampling fraction for each stratum regardless of the differences in population size of the strata. It is much like assembling a smaller population that is specific to the relative proportions of the subgroups within the population.

Disproportionate Stratified Random Sampling

The only difference between proportionate and disproportionate stratified random sampling is their sampling fractions. With disproportionate sampling, the different strata have different sampling fractions.

The precision of this design is highly dependent on the sampling fraction allocation of the researcher. If the researcher commits mistakes in allotting sampling fractions, a stratum may either be overrepresented or underrepresented which will result in skewed results.

4. this is also a random sampling method as it is remains unknown who will enter the middle school building in the morning and then the number of students who will return their completed surveys to the office at the end of the day.