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Chapter 1: Problem 29

In Problems 11-22, identify the type of sampling used. The mathematics department at a university wishes to administer a survey to a sample of students taking college algebra. The department is offering 32 sections of college algebra, similar in class size and makeup, with a total of 1280 students. They would like the sample size to be roughly \(10 \%\) of the population of college algebra students this semester. How might the department obtain a simple random sample? A stratified sample? A cluster sample? Which method do you think is best in this situation?

Short Answer

Expert verified
To obtain a representative sample in this situation, a stratified sample is the best method since it ensures equal representation across all sections.

Step by step solution

01

Identify the Population and Sample Size

The population consists of 1280 students enrolled in 32 sections of college algebra. The sample size should be roughly 10% of the population, so the department is looking to survey about 128 students.
02

Simple Random Sample

In a simple random sample, each student has an equal chance of being selected. The department can assign a unique number to each of the 1280 students and use a random number generator to select 128 unique numbers. The students corresponding to these numbers will be part of the sample.
03

Stratified Sample

In a stratified sample, the population is divided into subgroups (strata), and a sample is taken from each subgroup. For this scenario, each section of the algebra class can be considered a stratum. If the sections are similar in size and makeup, the department can randomly select students from each section in proportion to the number of students in that section. For instance, if each section has 40 students, they could select 4 students randomly from each of the 32 sections (40 students per section x 32 sections = 1280 total; 4 students per section x 32 sections = 128).
04

Cluster Sample

In a cluster sample, the population is divided into clusters, and then entire clusters are randomly selected. For this case, the department can consider each section as a cluster and then randomly select several sections. All students in the selected sections would be surveyed. If they need 128 students, they can randomly choose 4 sections since each section has approximately 40 students.
05

Evaluate the Best Method

Simple random sampling ensures every student has an equal chance of being selected but may be logistically complex. Stratified sampling ensures representation from all sections, which is useful if the sections have varied characteristics. Cluster sampling is logistically simpler as fewer sections are involved, but it may introduce bias if the chosen sections are not representative. Given that the sections are similar in size and make-up, a stratified sample would likely provide the most accurate and representative results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Random Sample
A simple random sample gives each individual in a population an equal chance of being selected. This method is akin to drawing names from a hat.
This ensures that every possible sample of a given size has the same chance of being chosen. To obtain a simple random sample in the exercise scenario, the mathematics department could:
  • Assign a unique number from 1 to 1280 to each student.
  • Use a random number generator to select 128 unique numbers from this range.
  • Survey the students whose numbers were selected.
This method helps avoid bias but can be logistically cumbersome when dealing with large populations.
Stratified Sample
A stratified sample divides the population into subgroups (strata), ensuring that every subgroup is represented in the sample. For the college algebra survey, each of the 32 sections can be considered a stratum.
First, the department could confirm that each section is similar in size and makeup. If each section has 40 students, they would then:
  • Randomly select 4 students from each of the 32 sections.
  • This way, all strata (sections) are proportionally represented.
Stratified sampling increases the precision of the sample estimate by ensuring each subgroup's characteristics are included. It鈥檚 particularly helpful when the population has distinct subgroups that might influence the study's outcome.
Cluster Sample
A cluster sample divides the population into clusters, then randomly selects entire clusters for the survey. These clusters should ideally be mini-representations of the population.
In the given scenario, the department can treat each algebra section as a cluster. To obtain a cluster sample, they could:
  • Randomly select 4 out of the 32 sections (since each has about 40 students).
  • Survey all students within these selected sections.
Cluster sampling is often easier to administer because it reduces the number of locations to visit. However, it might introduce bias if the chosen clusters are not representative of the entire population.
Educational Survey
Conducting an educational survey involves systematically collecting data to understand aspects of education. Surveys can cover a wide range of topics, such as student performance, teaching effectiveness, or curriculum satisfaction.
Key elements when planning a survey include:
  • Defining a clear objective: What does the department want to learn?
  • Choosing the right sampling method: simple random, stratified, or cluster sample.
  • Ensuring the sample is representative of the entire population.
  • Designing well-structured questions that are easy to understand.
  • Ensuring the confidentiality and anonymity of respondents.
In our exercise, the objective is to gather feedback from college algebra students. Given the 32 sections with similar size and makeup, ensuring a representative sample will help the department gather comprehensive insights.

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