To determine which estimate at a [tex]\( 95\% \)[/tex] confidence level most likely comes from a small sample, we need to assess the margin of error for each option provided. The margin of error gives us an indication of how precise the estimate is, and typically, a larger margin of error suggests a smaller sample size.
Let's analyze each option:
1. Option A: [tex]\( 71\% \pm 18\% \)[/tex]
- Margin of error: [tex]\( 18\% \)[/tex]
2. Option B: [tex]\( 60\% \pm 4\% \)[/tex]
- Margin of error: [tex]\( 4\% \)[/tex]
3. Option C: [tex]\( 65\% \pm 2\% \)[/tex]
- Margin of error: [tex]\( 2\% \)[/tex]
4. Option D: [tex]\( 62\% \pm 6\% \)[/tex]
- Margin of error: [tex]\( 6\% \)[/tex]
The margin of error is the variability allowed in the estimate. Generally, a larger margin of error indicates a higher level of uncertainty in the estimate. This uncertainty often results from analyzing a smaller sample size. Therefore, among the given options, the largest margin of error indicates which estimate likely comes from the smallest sample.
Comparing the margins of error:
- 18% (Option A)
- 4% (Option B)
- 2% (Option C)
- 6% (Option D)
The largest margin of error is 18%, which corresponds to Option A.
Thus, the estimate that most likely comes from a small sample is:
A. [tex]\( 71\% \pm 18\% \)[/tex]
This conclusion is based on recognizing that a larger margin of error implies a smaller sample size, which introduces greater variability and less precision into the estimate.
