Problem 13 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Surveys are mailed to every other household in a neighborhood. Each survey that is returned is used. Describe and correct the error in identifying the type of sample that is used. Because the surveys were mailed to every other household, the sample is a systematic sample.

Step-by-Step Solution

Answer

The error in the statement is that while the surveys are systematically distributed, the resulting sample is a convenience sample, as it only includes surveys that were returned.

Step 1: Identify Sampling Method

Identify and outline the type of sample used in the survey. In this case, the sampling method is described as a systematic sample, which takes every nth item from the population.

Step 2: Analyze the Statement

Analyze the statement to determine if there is indeed an error in the type of sample used. The key point is that the surveys are sent to every other household, indicating that selection isn't completely random, but systematic.

Step 3: Correct the Error

Determine if there is a need to correct the error. Where the statement might be misleading is that it doesn't clarify that use of the surveys depends on them being returned. Therefore, though the method for sending out the surveys is systematic, the sample actually used for the study is a convenience sample, as it is based solely on the surveys that are returned, not all that are sent out. The correct statement is: Because the surveys were mailed to every other household, the method for sending out the surveys is systematic, however the sample used is a convenience sample.

Key Concepts

Systematic Sampling

In the realm of survey methodology, systematic sampling is a technique where a sample is drawn from a larger population by selecting elements at a regular interval, such as every nth item. To begin a systematic sample, you would select a starting point at random and then pick every nth household, for instance.

This method helps to ensure that the sample is spread across the entire population, which can improve the representativeness of the results. It's important to note that while systematic sampling is more structured than simple random sampling, it still carries a risk of sampling error if periodic patterns occur in the population that align with the sampling interval.

Convenience Sample

Unlike systematic sampling, a convenience sample involves collecting data that is easiest to obtain. It is chosen based on what is immediately available and convenient to the researcher, often leading to a biased sample as it may not accurately represent the full diversity of the larger population.

In our textbook exercise, while the surveys were mailed out using a systematic approach, the sample ultimately relied on the willingness of recipients to respond, which is a hallmark of convenience sampling. This type of sampling requires less effort and is less costly, but it significantly increases the chance of sampling error and can limit the generalizability of the findings.

Survey Methodology

Survey methodology encompasses the techniques and strategies used in the design and execution of surveys. It covers everything from selecting a target population and sampling method to designing the questionnaire and analyzing the data collected.

Effective survey design requires careful planning to ensure the accuracy and reliability of the data. A well-crafted survey requires a sampling strategy that minimizes bias and leads to a representative sample. It's essential to clarify how participants are selected and how the data is gathered to avoid misconceptions about the type of sampling, as seen in our textbook exercise.

Sampling Error

Sampling error is the difference between the results obtained from a sample and the actual parameters of the entire population. This discrepancy arises due to the variations in the characteristics of individuals chosen for the sample compared to those not selected.

Every sampling method, including systematic and convenience sampling, is subject to sampling error. The size and randomness of the sample are critical in determining the magnitude of the sampling error. A larger, more randomly-selected sample typically reduces the error, leading to more confidence in the results of the survey. Understanding sampling error is key to evaluating the credibility and reliability of any survey's findings.