Answer :
To determine which estimate most likely comes from a small sample at a [tex]$95\%$[/tex] confidence level, we need to examine the width of the confidence intervals given for each option. The width of the confidence interval is directly related to the sample size: smaller sample sizes generally result in wider confidence intervals due to increased variability.
1. Option A: [tex]\(67\% (\pm 7\%)\)[/tex]
- The confidence interval for this estimate is [tex]\(67\% \pm 7\%\)[/tex], which translates to [tex]\([60\%, 74\%]\)[/tex].
- The width of this interval is [tex]\(74\% - 60\% = 14\%\)[/tex].
2. Option B: [tex]\(59\% (\pm 5\%)\)[/tex]
- The confidence interval for this estimate is [tex]\(59\% \pm 5\%\)[/tex], which translates to [tex]\([54\%, 64\%]\)[/tex].
- The width of this interval is [tex]\(64\% - 54\% = 10\%\)[/tex].
3. Option C: [tex]\(48\% (\pm 21\%)\)[/tex]
- The confidence interval for this estimate is [tex]\(48\% \pm 21\%\)[/tex], which translates to [tex]\([27\%, 69\%]\)[/tex].
- The width of this interval is [tex]\(69\% - 27\% = 42\%\)[/tex].
4. Option D: [tex]\(53\% (\pm 3\%)\)[/tex]
- The confidence interval for this estimate is [tex]\(53\% \pm 3\%\)[/tex], which translates to [tex]\([50\%, 56\%]\)[/tex].
- The width of this interval is [tex]\(56\% - 50\% = 6\%\)[/tex].
Out of all the provided options, Option C has the widest confidence interval ([tex]\(42\%\)[/tex]). This wide interval suggests a higher margin of error, which typically occurs in estimates derived from smaller samples.
Thus, the estimate at a [tex]$95\%$[/tex] confidence level that most likely comes from a small sample is:
Option C: [tex]\(48\% (\pm 21\%)\)[/tex].
1. Option A: [tex]\(67\% (\pm 7\%)\)[/tex]
- The confidence interval for this estimate is [tex]\(67\% \pm 7\%\)[/tex], which translates to [tex]\([60\%, 74\%]\)[/tex].
- The width of this interval is [tex]\(74\% - 60\% = 14\%\)[/tex].
2. Option B: [tex]\(59\% (\pm 5\%)\)[/tex]
- The confidence interval for this estimate is [tex]\(59\% \pm 5\%\)[/tex], which translates to [tex]\([54\%, 64\%]\)[/tex].
- The width of this interval is [tex]\(64\% - 54\% = 10\%\)[/tex].
3. Option C: [tex]\(48\% (\pm 21\%)\)[/tex]
- The confidence interval for this estimate is [tex]\(48\% \pm 21\%\)[/tex], which translates to [tex]\([27\%, 69\%]\)[/tex].
- The width of this interval is [tex]\(69\% - 27\% = 42\%\)[/tex].
4. Option D: [tex]\(53\% (\pm 3\%)\)[/tex]
- The confidence interval for this estimate is [tex]\(53\% \pm 3\%\)[/tex], which translates to [tex]\([50\%, 56\%]\)[/tex].
- The width of this interval is [tex]\(56\% - 50\% = 6\%\)[/tex].
Out of all the provided options, Option C has the widest confidence interval ([tex]\(42\%\)[/tex]). This wide interval suggests a higher margin of error, which typically occurs in estimates derived from smaller samples.
Thus, the estimate at a [tex]$95\%$[/tex] confidence level that most likely comes from a small sample is:
Option C: [tex]\(48\% (\pm 21\%)\)[/tex].
