Question

use the confidence interval to find the estimated margin of error. then find the sample mean.
a biologist reports a confidence interval of (2.8,4.0) when estimating the mean height (in centimeters) of a sample of seedlings.
the estimated margin of error is
the sample mean is

Explanation:

Step1: Calculate the margin of error

The formula for the margin of error \(E\) using a confidence interval \((a,b)\) is \(E=\frac{b - a}{2}\).
Here \(a = 2.8\) and \(b=4.0\), so \(E=\frac{4.0 - 2.8}{2}=\frac{1.2}{2}\).

Step2: Calculate the sample mean

The formula for the sample mean \(\bar{x}\) using a confidence interval \((a,b)\) is \(\bar{x}=\frac{a + b}{2}\).
Substitute \(a = 2.8\) and \(b = 4.0\) into the formula: \(\bar{x}=\frac{2.8+4.0}{2}=\frac{6.8}{2}\).

Answer:

The estimated margin of error is \(0.6\).
The sample mean is \(3.4\).