Question

What survey design is used in each of these situations? a. A random sample of \(n=50\) city blocks is selected, and a census is done for each single-family dwelling on each block. b. The highway patrol stops every 10th vehicle on a given city artery between 9: 00 A.M. and 3: 00 P.M. to perform a routine traffic safety check. c. One hundred households in each of four city wards are surveyed concerning a pending city tax relief referendum. d. Every 10 th tree in a managed slash pine plantation is checked for pine needle borer infestation. e. A random sample of \(n=1000\) taxpayers from the city of San Bernardino is selected by the Internal Revenue Service and their tax returns are audited.

Short answer

a) Simple Random Sampling b) Systematic Sampling c) Stratified Sampling d) Cluster Sampling

Step-by-step solution

Scenario a)

This is an example of cluster sampling. The city blocks are clusters and a random sample of 50 clusters (city blocks) is chosen. b. The highway patrol stops every 10th vehicle on a given city artery between 9:00 A.M. and 3:00 P.M. to perform a routine traffic safety check.

Scenario b)

This is an example of systematic sampling. Every 10th vehicle is stopped, meaning there is a fixed interval between the selected samples. c. One hundred households in each of four city wards are surveyed concerning a pending city tax relief referendum.

Scenario c)

This is an example of stratified sampling. The city wards serve as strata, with a random sample of 100 households taken from each stratum. d. Every 10 th tree in a managed slash pine plantation is checked for pine needle borer infestation.

Scenario d)

This is an example of systematic sampling. Similar to scenario b, every 10th tree is checked, indicating a fixed interval between the selected trees. e. A random sample of \(n=1000\) taxpayers from the city of San Bernardino is selected by the Internal Revenue Service and their tax returns are audited.

Scenario e)

This is an example of simple random sampling, as the problem states that a random sample is selected, suggesting that each taxpayer in the city has an equal chance of being chosen for the audit.

Key concepts

Cluster Sampling

Cluster sampling is a method used when it's more efficient or practical to divide a population into clusters, or groups, before sampling. Instead of selecting individuals directly, entire clusters are randomly chosen. This approach is especially useful when dealing with large populations spread over a wide area.
In the example provided from the exercise, city blocks serve as the clusters. By conducting a full census on each selected block, researchers can efficiently gather data from numerous individual households. This technique is beneficial when clusters are naturally occurring, like geographical areas, and it's unfeasible to list every member of a large population for sampling.
The advantage of cluster sampling is in cost and time savings, as it requires fewer logistical resources. However, it's crucial that the clusters themselves are randomly chosen; otherwise, it might lead to biased outcomes. The key here is that while the cluster is randomly selected, all members of the selected cluster are included in the sample.

Systematic Sampling

Systematic sampling involves selecting units from a population at regular intervals. It is both simple and convenient. To apply this, a researcher first determines a fixed interval, known as the sampling interval. They then randomly select a starting point and continue to select at every nth interval.
This method is illustrated in the exercise when every 10th vehicle or tree is chosen for observation. Systematic sampling is especially useful when a complete list of the population is available, and the population is not ordered in a way that hides periodic patterns. This ensures that the sample is evenly spread across the population, which can be more representative than a random sample.
However, care must be taken to ensure that the starting point and interval don't coincide with any hidden periodic patterns in the original list, as this could introduce sampling bias. If done correctly, systematic sampling offers a straightforward method for data collection with minimal researcher bias.

Stratified Sampling

Stratified sampling is designed to ensure that different subgroups within a population are adequately represented in the sample. The population is divided into homogeneous subgroups, or strata, and samples are drawn from each stratum.
In the given scenario from the exercise, the city wards are used as strata. By sampling 100 households from each ward, the survey accounts for any variability between the wards. This sampling method is particularly beneficial when there is a known diversity within the population that could influence the research outcomes. By making sure that each stratum is represented, the method provides more accurate and reliable results.
The main advantage of stratified sampling is its capacity for more precise estimates if the variability within each stratum is smaller than between them. The challenge, however, lies in correctly defining the strata and ensuring the samples from each are representative of that group. Once this is achieved, stratified sampling can significantly improve the reliability of the results.

Simple Random Sampling

Simple random sampling is one of the most straightforward methods. Every member of a population has an equal chance of being selected. This can be achieved using random numbers to pick individuals or through drawing lots.
The example in the exercise involving the IRS auditing a random sample of taxpayers demonstrates this method. Each taxpayer has an equal opportunity to be part of the sample, making this method unbiased and fair.
While simple random sampling is conceptually easy, it can sometimes be impractical for very large populations where creating a complete list is challenging. Additionally, it may not be the most efficient method if there is considerable variability within the population, as more nuanced methods like stratified sampling might provide better insight.
Nonetheless, when practical, simple random sampling is an excellent choice for ensuring each member of the population is equally represented, leading to valid and generalizable findings.