Question

Suppose that you were asked to help design a survey of adult city residents in order to estimate the proportion that would support a sales tax increase. The plan is to use a stratified random sample, and three stratification schemes have been proposed. Scheme 1: Stratify adult residents into four strata based on the first letter of their last name (A-G, \(\mathrm{H}-\mathrm{N}, \mathrm{O}-\mathrm{T}, \mathrm{U}-\mathrm{Z})\) Scheme 2: Stratify adult residents into three strata: college students, nonstudents who work full time, nonstudents who do not work full time. Scheme 3: Stratify adult residents into five strata by randomly assigning residents into one of the five strata. Which of the three stratification schemes would be best in this situation? Explain.

Short answer

The second scheme would be the best in this situation because it stratifies residents based on their socioeconomic status, a factor likely to affect their opinion on a tax increase. Thus, this scheme is more likely to create strata that are internally homogeneous and externally heterogeneous, making it effective for the stated goal of the survey.

Step-by-step solution

Analyzing scheme 1

In the first scheme, residents are stratified according to the first letter of their last name. This scheme might not be effective because the last names of residents are not related to their opinion on a tax increase. Thus, it wouldn't make the strata internally homogeneous with respect to the survey's subject.

Analyzing scheme 2

The second scheme stratifies residents based on whether they are college students, full-time workers, or non-full-time workers. These are socioeconomic groups, and economic condition can relate to one's opinion on a tax increase. Therefore it is more likely that this scheme would create strata that are internally homogeneous and externally heterogeneous, making it more effective than the first scheme.

Analyzing scheme 3

In the third scheme, residents are randomly assigned to one of five strata. This doesn't seem any better than simple random sampling and wouldn't ensure that each stratum is internally homogeneous regarding the survey's subject. Thus, this scheme might be ineffective.

Key concepts

Survey Design

Survey design is a crucial aspect of conducting research as it directly impacts the quality of the data collected. When designing a survey to estimate the proportion of adult city residents who would support a sales tax increase, careful consideration must be given to the scope of the survey, the target population, and the methodology used for gathering information. A well-designed survey will minimize biases and provide a representative snapshot of the population's views.

In the proposed exercise scenario, the survey design must ensure that the sample reflects the diverse opinions of the city's adult population regarding the proposed sales tax increase. An excellent survey design includes clear and unbiased question wording, a reasonable method of contacting respondents, and a sampling approach that accurately reflects the population's characteristics. Incorporating stratified random sampling, for example, can improve the reliability of the survey results by ensuring subgroups within the population are proportionally represented in the sample.

Sampling Strategies

Sampling strategies are methods used to select a subset of individuals from a population to participate in a survey. The goal is to make inferences about the entire population based on the responses of this subset. There are various sampling strategies, including simple random sampling, systematic sampling, clustered sampling, and stratified random sampling.

Stratified Random Sampling

Stratified random sampling is particularly efficient when the population is heterogeneous. This method involves dividing the population into strata, or groups, based on shared characteristics before sampling. The exercise suggests three different schemes for creating these strata. Scheme 1 sorts individuals by their last name, which has no obvious connection to their tax opinions and would not create strata that accurately reflect the survey's purpose. Scheme 2 takes into account socioeconomic characteristics, potentially offering a more nuanced view of tax support across different economic groups. Lastly, Scheme 3 randomly assigns people to strata, which does not improve upon simple random sampling and may not produce representative strata. Among these strategies, Scheme 2 seems to hold the most promise due to the relationship between economic status and opinions on tax changes.

Tax Increase Opinion Survey

A tax increase opinion survey aims to gauge the public's acceptance or rejection of a proposed increment in tax rates, in this case, a sales tax increase. When conducting such a survey, it is essential to tap into the various opinions that exist within a community. Different demographics may have significantly disparate views on taxation based on their income, employment status, education level, and other societal factors.

The exercise's scheme 2, which proposes stratification by college students, full-time workers, and non-full-time workers, targets these diverse economic segments. This approach enhances the likelihood of capturing the varying perspectives on a sales tax increase that exist within a city. By analyzing the second scheme's potential to produce a sampling design that is both internally homogeneous within strata and externally heterogeneous across them, it becomes clear that scheme 2 is the most targeted and relevant strategy for a tax increase opinion survey.