Question

give an example of how you would create a sample from the population of the united states using cluster sampling and stratified sampling. then, compare and contrast these two methods.

Explanation:

Cluster Sampling Example:

1. Divide into Clusters: Consider the population of the US. We can divide the entire country into clusters based on geographical regions (e.g., states or counties). Let's say we use counties as clusters.
2. Randomly Select Clusters: Randomly select a certain number of counties (e.g., 50 counties) from all the counties in the US.
3. Sample from Selected Clusters: Include all the individuals (or a census) from the selected 50 counties in our sample. So, if a county has a population of 10,000, all 10,000 people from each of the 50 selected counties are part of the sample.

Stratified Sampling Example:

1. Define Strata: Define strata based on a characteristic, such as income levels (e.g., low - income, middle - income, high - income) or age groups (e.g., 0 - 18, 19 - 64, 65+). Let's use income levels as strata.
2. Proportionate Sampling within Strata: Determine the proportion of each income stratum in the overall US population. If, for example, 30% of the US population is in the low - income stratum, 50% in the middle - income, and 20% in the high - income, we then take a random sample from each stratum such that the sample size from each stratum is proportional to its size in the population. So, if we want a total sample size of 1000, we would take 300 (30% of 1000) from the low - income stratum, 500 (50% of 1000) from the middle - income, and 200 (20% of 1000) from the high - income stratum, with each of these samples being randomly selected from their respective strata.

Compare and Contrast:

Similarities:

• Both are probability sampling methods, meaning that each element in the population has a known (or at least calculable) chance of being selected.
• Both aim to create a representative sample of the population, although they do so in different ways.

Differences:

• Cluster Sampling:
• Clusters are heterogeneous within themselves (they should be as similar as possible to the overall population in terms of the characteristics we are interested in) and homogeneous between clusters (different clusters are similar to each other). We sample entire clusters.
• It is often more cost - effective and convenient, especially for large geographical areas, as we can focus our sampling efforts on a few clusters rather than the entire population. However, if the clusters are not representative, the sample may be biased.
• Stratified Sampling:
• Strata are homogeneous within themselves (elements within a stratum share a common characteristic) and heterogeneous between strata (different strata have distinct characteristics). We sample from each stratum.
• It is more likely to produce a representative sample, especially when the population has distinct subgroups (strata) that we want to ensure are represented in the sample. It can also provide more precise estimates (lower sampling error) if the strata are well - defined and the variable of interest varies more between strata than within strata.

Answer:

Cluster Sampling Example:

1. Divide the US into county - level clusters.
2. Randomly select 50 counties.
3. Include all people from the selected 50 counties in the sample.

Stratified Sampling Example:

1. Define income - based strata (low, middle, high - income).
2. Take a proportionate random sample from each stratum (e.g., 300 from low - income, 500 from middle - income, 200 from high - income for a sample of 1000).

Compare and Contrast:

• Similarities: Both are probability sampling methods aiming for representativeness.
• Differences:
• Cluster: Heterogeneous within clusters, sample entire clusters, cost - effective but risk of bias.
• Stratified: Homogeneous within strata, sample from each stratum, more representative and precise.