Question: 5. (6 points) A poll pertaining to environmental concerns had the following footnote: “Based on a…

5. (6 points) A poll pertaining to environmental concerns had the following footnote: “Based on a sample of 1,763 adults, the margin of error is plus or minus 2.5 percentage points.” Find the level of confidence of the poll.

Solution

To find the level of confidence for a poll with a given margin of error and sample size, typically the formula used is: \[ \text{Margin of Error} = z \times \sqrt{\frac{p(1-p)}{n}} \] where: - \(z\) is the z-score corresponding to the desired confidence level, - \(p\) is the estimated proportion (often assumed to be 0.5 for maximum variability if not given), - \(n\) is the sample size. Step 1: Identify the given values - Margin of Error = 2.5% = 0.025 - Sample size \(n = 1763\) Step 2: Assume \(p = 0.5\) (since it is not provided) Step 3: Rearrange the formula to solve for \(z\): \[ z = \frac{\text{Margin of Error}}{\sqrt{\frac{p(1-p)}{n}}} \] Step 4: Substitute the known values into the equation: \[ z = \frac{0.025}{\sqrt{\frac{0.5 \times 0.5}{1763}}} \] Step 5: Calculate the denominator: \[ \sqrt{\frac{0.25}{1763}} = \sqrt{0.0001418} \approx 0.0119 \] Step 6: Calculate \(z\): \[ z = \frac{0.025}{0.0119} \approx 2.10 \] Step 7: Determine the z-score’s confidence level A z-score of approximately 2.10 corresponds to a confidence level close to 96%. Therefore, the confidence level of the poll is approximately 96%.