To find the margin of error for the sample survey, let's break down the solution step by step using the given information.
- The sample size in this problem is [tex]\( 800 \)[/tex].
- The total number of students surveyed who want the mascot to change is [tex]\( 250 \)[/tex].
First, estimate the population proportion, denoted as [tex]\(\hat{p}\)[/tex]:
[tex]\[
\hat{p} = \frac{250}{800} = 0.3125
\][/tex]
Next, calculate [tex]\(1 - \hat{\rho}\)[/tex]:
[tex]\[
1 - \hat{p} = 1 - 0.3125 = 0.6875
\][/tex]
We are given the [tex]\( z \)[/tex]-score for a [tex]\( 99\% \)[/tex] confidence level, which is [tex]\( 2.58 \)[/tex].
Now, we can calculate the margin of error [tex]\( E \)[/tex] using the formula:
[tex]\[
E = z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\][/tex]
Substitute the values:
[tex]\[
E = 2.58 \cdot \sqrt{\frac{0.3125 \cdot 0.6875}{800}}
\][/tex]
Computing this, we get the margin of error. When rounded to the nearest tenth percent, the result is [tex]\( 4.2\% \)[/tex].
Therefore, the complete statements are:
- The sample size in this problem is [tex]\( 800 \)[/tex].
- Estimate the population proportion as [tex]\( 0.3125 \)[/tex].
- [tex]\( (1 - \hat{\rho}) = 0.6875 \)[/tex].
- When the margin of error is calculated using the formula [tex]\( E = z^* \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \)[/tex], to the nearest tenth percent, the result is [tex]\( 4.2 \% \)[/tex].
