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Algebra and Data Analysis - Instructional Strategies

Algebra/ Data Analysis Test Clarification on Sampling

The purpose of this document is to clarify the concepts of sampling that are addressed in the Mathematics High School Core Learning Goals indicator 3.1.1: The student will design and/or conduct an investigation that uses statistical methods to analyze data and communicate results. The assessment limits identify the types of investigations that may be assessed on the HSA as simple random sampling (the concept clarified in this document) and probability simulations.

Simple random sampling, probability sampling, stratified random sampling, systematic sampling and multistage sampling are some of the legitimate sampling methods that are widely used depending on what is being measured. The high school assessment will only test students on their knowledge of simple random sampling. Simple random sampling is defined as sampling a group of n individuals from the population in such a way that every combination of n individuals has an equal chance of being the sample selected. To obtain a simple random sample, each member of the population must be equally likely to be chosen and the members of the sample must be chosen independently of each other. This is the definition of simple random sampling given in the CLG.

The following information describes a few of the ways sampling may be addressed on the HSA.

  1. The following is a 2000 public released item in which students are asked to evaluate different methods of sampling to determine if they are examples of simple random sampling.
    A doctor wants to conduct a survey using a random sample of her 1,500 patients. Below are three methods she is considering.
    Method 1: pick every fourth patient that enters the doctor's waiting room on a randomly selected day
    Method 2: number the doctor's patients from 1 to 1,500 and then generate random numbers (ignoring repeats) to select 30 patients
    Method 3: select every 20th patient's folder from the filing cabinet drawers until 30 patient names are chosen

    Complete the following in the Answer Book:

    • Which method should the doctor choose? Use principles of simple random sampling to justify your answer.
    • Use principles of simple random sampling to justify why she should not choose the other two methods.

    Method 2 is an example of simple random sampling. Each patient has the same chance of being chosen (all patients are assigned exactly one number) and the patients are chosen independently of one another (the appearance of one random number does not affect what random number will occur next).

    Method 1 is not an example of simple random sampling because many of the doctor's patients will not be in the waiting room that day and therefore do not have an equal chance of being chosen. Also, the patients are not chosen independently of each other because once the fourth patient is surveyed the fifth patient does not have a chance to be selected. Method 3 also is not an example of simple random sampling because patients whose folders are near the end of the filing cabinet drawers do not have a chance to be selected and for the same reason as Method 1 patients are not chosen independently of one another. With a few modifications, Method 3 could be an example of systematic sampling which is a method for obtaining a random sample. Therefore, one must be careful when saying one method is more random than another method.

    A complete response to an item of this type must state how both principles of simple random sampling are met with the correct method and at least one principle that is not met for each of the two incorrect methods. Many students will not select method 2 because they think it is difficult to carry out. The adjective of simple has to do with the method being a simple design; it does not describe how easy it is to carry out. It is also important to note that using a simple random sampling method does not guarantee a representative sample.

  2. The HSA may ask students to design an investigation. Given a scenario, students may be cued: Based on simple random sample principles, explain the two key elements that should be part of the sampling design. Or students may be asked to design a simple random sample and use mathematics to justify their answer. An exemplary student response should address both the fact that each member of the population is equally likely to be chosen and that the members of the sample are chosen independently of each other.
  3. The assessment limits for 3.1.1 also identify other important concepts that relate to sampling. Students taking the HSA must have knowledge of the concepts of representative samples, bias, and sample size. These concepts may be assessed in any of the HSA item types.
    • A representative sample is a sample that reflects the characteristics of the population. The method of selecting a sample does not exclude, does not over represent or does not under represent any part of the population.
    • If a method of selecting a sample tends to over represent or to under represent some part of the population, the method is biased and the resulting samples tend not to be representative of the population.
    • Sample size is important when conducting a survey since having an appropriate sample size gives more confidence in the results. Larger samples give more accurate results than smaller samples.
Instructional Strategies