Question

use the given confidence interval to find the margin of error and the sample proportion. (0.633, 0.661) e = □ (type an integer or a decimal.) hat { p } = □ (type an integer or a decimal.)

Explanation:

Step1: Find the sample proportion

The sample proportion $\hat{p}$ is the mid - point of the confidence interval.
The formula for the mid - point of an interval $(a,b)$ is $\hat{p}=\frac{a + b}{2}$.
Here, $a = 0.633$ and $b=0.661$.
So, $\hat{p}=\frac{0.633+0.661}{2}=\frac{1.294}{2}=0.647$.

Step2: Find the margin of error

The margin of error $E$ is half of the length of the confidence interval.
The length of the confidence interval is $b - a$.
So, $E=\frac{b - a}{2}$.
Substitute $a = 0.633$ and $b = 0.661$ into the formula:
$E=\frac{0.661-0.633}{2}=\frac{0.028}{2}=0.014$.

Answer:

$E = 0.014$; $\hat{p}=0.647$