Which of the following estimates at a [tex][tex]$95 \%$[/tex][/tex] confidence level most likely comes from a small sample?

A. [tex][tex]$71 \%( \pm 18 \%)$[/tex][/tex]
B. [tex][tex]$60 \%( \pm 4 \%)$[/tex][/tex]
C. [tex][tex]$65 \%( \pm 2 \%)$[/tex][/tex]
D. [tex][tex]$62 \%( \pm 6 \%)$[/tex][/tex]



Answer :

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.