The school board in a certain school district obtained a random sample of $200$residents and asked if they were in favor of raising property taxes to fund the hiring of more statistics teachers. The resulting confidence interval for the true proportion of residents in favor of raising taxes was $(0.183,0.257)$. Which of the following is the margin of error for this confidence interval?

a. $0.037$

b. $0.074$

c. $0.183$

d. $0.220$

e.$0.257$

Given in the question that, the school board in a certain school district obtained a random sample of $200$residents and asked if they were in favor of raising property taxes to fund the hiring of more statistics teachers. The resulting confidence interval for the true proportion of residents in favor of raising taxes was $(0.183,0.257)$.We need to find the margin of error for this confidence interval.

Remember that the confidence interval for proportion has the shape shown on the left; we'll use this to figure out the margin of error. The form to the left is obtained by subtracting the second value from the first value. As you can see, we'll take the second value in the confidence interval, subtract the first value, and divide by two to find the margin of error. As a result,

$(\hat{p}-MOE,\hat{p}+MOE)$

role="math" localid="1654232673443" $$\begin{gathered} \Rightarrow (\hat{p}+MOE)-(\hat{p}-MOE) \\ =2\times MOE \end{gathered}$$

$\Rightarrow MOE=\frac{0.257-0.183}{2}$

$=0.037$

As a result, option (a) is the proper choice.