when 1376 voters were polled, 82% said they were voting \yes\ on an initiative measure. find the margin of error. ±37.1% ±27% ±2.7% ±3.7% question 6 2 pts when 1376 voters were polled, 82% said they were voting \yes\ on an initiative measure. if the margin of error is ±2.7% what is the confidence interval? 79.3% - 84.7% 78.3% - 85.7% 55% - 100% 44.9% - 100%

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Accepted Answer

# Explanation:
## Step1: Calculate the margin of error
The formula for the margin of error $E$ for a proportion in a simple random sample is $E = z\sqrt{\frac{p(1 - p)}{n}}$. For a large - sample (when $np\geq5$ and $n(1 - p)\geq5$, here $n = 1376$, $p=0.82$, $np=1376\times0.82 = 1128.32\geq5$, $n(1 - p)=1376\times(1 - 0.82)=247.68\geq5$) and a common confidence level (assuming 95% confidence level, $z\approx2$):

$$
\begin{align*}
E&=2\sqrt{\frac{0.82\times(1 - 0.82)}{1376}}\
&=2\sqrt{\frac{0.82\times0.18}{1376}}\
&=2\sqrt{\frac{0.1476}{1376}}\
&=2\sqrt{0.00010727}\
&=2\times0.01036\
&\approx 0.027 = 2.7\%
\end{align*}
$$

## Step2: Calculate the confidence interval
The confidence interval for a proportion is given by $\hat{p}-E<p<\hat{p} + E$, where $\hat{p}=0.82$ (or $82\%$) and $E = 0.027$ (or $2.7\%$)

$$
\begin{align*}
\hat{p}-E&=82\% - 2.7\%=79.3\%\
\hat{p}+E&=82\%+ 2.7\%=84.7\%
\end{align*}
$$

# Answer:
For the margin of error question: $\pm2.7\%$
For the confidence interval question: $79.3\%-84.7\%$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the margin of error

Step2: Calculate the confidence interval

Answer: