Question

For their class project, a group of Statistics students decide to survey the student body to assess opinions about the proposed new student center. Their sample of 200 contained 50 first-year students, 50 sophomores, 50 juniors, and 50 seniors. a) Do you think the group was using an SRS? Why? b) What sampling design do you think they used?

Short answer

a) No, because it divides students by year. b) Stratified sampling.

Step-by-step solution

Understanding Simple Random Sample (SRS)

A Simple Random Sample (SRS) is a sampling method where every individual in the population has an equal chance of being selected. In an SRS of size 200 from a student body, any group of 200 students could potentially be chosen, regardless of their year.

Analyzing the Given Sample

The sample of 200 students is divided evenly among four distinct groups: 50 first-year students, 50 sophomores, 50 juniors, and 50 seniors. The even distribution among these groups indicates purposeful selection based on the year of study.

Determining if SRS was Used

Since each group (first-year, sophomore, junior, senior) contains exactly 50 students, the sample does not reflect a scenario where all students were equally likely to be chosen, as would be the case in an SRS. Thus, the group was not using an SRS.

Identifying the Actual Sampling Design

The sample reflects a stratified sampling design. In stratified sampling, the population is divided into subgroups (strata) that share similar characteristics (in this case, the year of study). Samples are drawn from each subgroup, often in equal numbers or proportionate to the population size.

Key concepts

Simple Random Sample (SRS)

Simple Random Sample, or SRS, is a technique used to select a sample from a population where every individual has an equal chance of being picked. Imagine you have a giant raffle drum with everyone's name on a ticket. Every time you draw a name, it's completely random and nobody knows who will be chosen next. This is how an SRS works. In practical terms, if you were to select 200 students out of a whole student body at random, every combination of 200 individuals would have the same chance of being selected. It's like flipping a coin for every student: total fairness and unpredictability in who is chosen. This method is very straightforward, but it requires careful management to ensure randomness. It's excellent for unbiased results as there's no grouping or sorting in the selection process. However, it's not always efficient, especially if the population is large or diverse, because you might miss out on capturing every segment of the population.

Stratified Sampling

Stratified Sampling is a well-structured sampling method where the population is first divided into distinct segments or "strata." These strata are groups of individuals that share common characteristics important to the research, such as class year, age, income levels, etc. The main advantage of stratified sampling is the ability to ensure that each subgroup is adequately represented. In our example, the student body was divided into first-year students, sophomores, juniors, and seniors. From each of these groups, a sample of 50 students was taken. This ensures every class year is represented equally in opinions about the new student center. This method can provide more precise and reliable results than an SRS, especially if the subgroups have distinct attributes that influence the research outcomes. Stratified sampling is particularly beneficial in studies where it's vital to ensure smaller or varying subgroups are not left out of the sample.

Population

The term Population in statistics refers to the entire group that you're interested in studying. It's not limited to people; it can include plants, animals, events, or objects—essentially any set of entities that a researcher intends to analyze. For the statistics students, the population is the entire student body whom they want to survey about the new student center. Understanding who makes up the population is crucial before designing a sampling plan because it guides the selection method and sample size. In sampling exercises, defining your population clearly helps ensure that your sample accurately reflects the group you're studying. This step is essential to achieve meaningful results that can be confidently applied back to the population. By accurately representing their views or characteristics, conclusions drawn from the sample can be effectively generalized to the whole population.