Read the design before the numbers
The weighting task assumes positive random samples within two strata. The group task describes which branches participate. The distribution task uses a fixed finite population and uniform size-two subsets. Do not import an independent-draw formula into every design.
Explain what makes each answer different
For each task identify either the target population shares, the first-stage selection units or the distribution of the estimator. The answer explanations include the reason each alternative fails.
Try the questions
Population groups contain 750 and 250 units. Each contributes 20 sampled units with group means 4 and 12. What is the stratified population mean estimate?
- A.
8
- B.
6
- C.
16
Answer and explanation
Answer: B
Combine within-group sample means using the defined population-unit shares.
- A
This uses equal sample shares rather than the unequal population shares.
- B
Population shares 0.75 and 0.25 give 0.75 × 4 + 0.25 × 12 = 6.
- C
Adding group means does not form a population mean.
Two of eight branches are selected randomly, then ten records are selected within each chosen branch. Which description fits?
- A.
Two-stage cluster sampling
- B.
Sampling within every branch as a stratum
- C.
A census of every record
Answer and explanation
Answer: A
Both the group selection and within-group selection are explicit random stages.
- A
Branches are selected first, then units within selected branches.
- B
The described design leaves six branches unselected.
- C
Only selected records in selected branches enter.
Uniform size-two subsets are drawn from values 2,4,6,8. One observed sample mean is 3. What is the expectation of the sample mean over this design?
- A.
3
- B.
5
- C.
8
Answer and explanation
Answer: B
The unbiased simple-random estimator is centered at the finite population mean 5.
- A
That is one realized sample mean, not the center over all allowed samples.
- B
The six sample means 3,4,5,5,6,7 average to 5.
- C
The largest population value does not determine the expected mean.