Question

one day, a poll appeared on a certain website. it asked do you believe that rain can soothe the heart and cleanse the mind? unsurprisingly, 66% of 474 answers were yes. calculate the margin of error for this poll. round the percentage to one decimal place. do not round intermediate steps. if 474 people were asked that question and 66% of them replied yes, the margin of error is \boxed{}%

Explanation:

Step1: Recall the formula for margin of error

The formula for margin of error \(E\) for a proportion is \(E = z\sqrt{\frac{p(1 - p)}{n}}\). For a simple - random sample (assuming this poll is a simple - random sample approximation), when no confidence level is specified, we often use a \(95\%\) confidence level. For a \(95\%\) confidence level, \(z\approx1.96\). Here, \(p = 0.66\) (the proportion of "Yes" answers) and \(n = 474\) (the sample size).

Step2: Calculate \(1 - p\)

\(1 - p=1 - 0.66 = 0.34\)

Step3: Calculate the value inside the square root

\(\frac{p(1 - p)}{n}=\frac{0.66\times0.34}{474}=\frac{0.2244}{474}\approx0.0004734\)

Step4: Calculate the square root

\(\sqrt{\frac{0.66\times0.34}{474}}\approx\sqrt{0.0004734}\approx0.02176\)

Step5: Calculate the margin of error

\(E = 1.96\times\sqrt{\frac{0.66\times0.34}{474}}\approx1.96\times0.02176\approx0.043\)
To convert this to a percentage, we multiply by \(100\). So \(E\approx4.3\%\)

Answer:

\(4.3\)