Margin Of Error C72C7F

1. **State the problem:** We are asked to explain what it means when a poll result of 51% approval has a margin of error of plus or minus 3.5 percentage points. 2. **Understanding margin of error:** The margin of error indicates the range within which the true population parameter is expected to lie with a certain level of confidence (usually 95%). 3. **Formula for confidence interval:** The confidence interval for the true approval percentage $p$ is given by: $$\text{Confidence Interval} = \hat{p} \pm \text{Margin of Error}$$ where $\hat{p}$ is the sample proportion (here 51%). 4. **Apply the values:** $$51\% \pm 3.5\% = (51 - 3.5)\% \text{ to } (51 + 3.5)\% = 47.5\% \text{ to } 54.5\%$$ 5. **Interpretation:** This means that based on the sample of 1144 Americans, we can be confident that the true percentage of all Americans who approve of Brett Kavanaugh is between 47.5% and 54.5%. 6. **Why margin of error exists:** It accounts for sampling variability — different samples might give slightly different results, so the margin of error quantifies this uncertainty. 7. **Summary:** The statement "the margin of error was given as plus or minus 3.5 percentage points" means the reported 51% approval could realistically be as low as 47.5% or as high as 54.5% in the entire population, considering the sample size and variability.