A warehouse stores thousands of marble slabs of different shapes, sizes, and materi...
Check the final answer first, then review the worked steps.
Problem
A warehouse stores thousands of marble slabs of different shapes, sizes, and materials that are used for construction projects. The manager wants to estimate the average weight of these slabs. The slabs are arranged in stacks that have about 10 slabs of different varieties. The manager wants to weigh a random sample of 100 slabs. Why might the manager choose a cluster random sample instead of a stratified random sample in this setting? Choose all answers that apply:
Answer
A, C
Step-by-step solution
- Understanding Sampling Methods: The problem asks why a cluster random sample might be preferred over a stratified random sample in the context of weighing marble slabs stored in stacks.
- Stratified Random Sampling: This method involves dividing the population into subgroups (strata) based on a characteristic (e.g., material, size) and then taking a random sample from each stratum. This ensures representation from all subgroups.
- Cluster Random Sampling: This method involves dividing the population into clusters (e.g., stacks of slabs). Then, a random sample of clusters is selected, and all individuals within the selected clusters are included in the sample. Alternatively, a random sample of clusters can be selected, and then a random sample of individuals within those clusters is taken.
- Analyzing the Scenario: The marble slabs are arranged in stacks. The manager wants to estimate the average weight of 100 slabs.
3. Evaluating Option A: "It will probably be easier to weigh heavy slabs that are stacked together rather than pulling out some slabs from different stacks."
* In a cluster sample, the manager might select entire stacks. If the slabs within a stack are similar in weight (or if weighing them together is more convenient), this approach could be more efficient than randomly selecting individual slabs from many different stacks, which would require moving around the warehouse more. This makes option A a plausible reason.
4. Evaluating Option B: "A cluster sample chooses some slabs from each stack, so the manager will be more likely to get a representative sample."
This statement is incorrect. A cluster sample typically selects entire clusters (stacks) or a random sample of individuals within selected clusters. It does not necessarily choose some slabs from each* stack. Stratified sampling is designed to ensure representation from all subgroups (which could be defined by stack, or by slab characteristics within stacks), making it more likely to yield a representative sample across different types of slabs if strata are chosen appropriately. Therefore, option B is not a valid reason to prefer cluster sampling.
5. Evaluating Option C: "Each stack has a variety of slabs, so choosing all of the slabs in some of the stacks should represent the population fairly well."
* This statement suggests that the stacks themselves are diverse enough internally that selecting a few complete stacks (clusters) could provide a reasonable estimate of the overall population of slabs. If each stack contains a mix of slab types, sizes, and weights, then sampling entire stacks could indeed give a fair representation of the population's average weight, especially if the stacks are relatively homogeneous with respect to the characteristic being measured (weight) across different stacks. This makes option C a plausible reason.
- Conclusion: Based on the analysis, options A and C provide valid reasons why a cluster random sample might be chosen over a stratified random sample in this scenario due to potential efficiency and the assumption that stacks can serve as representative clusters.