Sure, let's go through the solution step-by-step:
1. Calculate the Sample Mean:
To find the sample mean, sum the heights and divide by the number of students.
Given heights: 60, 72, 68, 66, 73, 70
[tex]\[
\bar{x} = \frac{60 + 72 + 68 + 66 + 73 + 70}{6} = \frac{409}{6} = 68.16666666666667
\][/tex]
2. Estimate the Margin of Error:
To calculate the margin of error at a 95% confidence level, we need to use the z-score corresponding to the 95% confidence level, which is approximately 1.96. Also required is the population standard deviation (σ), which is given as 3 inches, and the sample size (n), which is 6.
The formula for the margin of error [tex]\(E\)[/tex] is:
[tex]\[
E = z \cdot \frac{\sigma}{\sqrt{n}}
\][/tex]
Plug in the values:
[tex]\[
E = 1.96 \cdot \frac{3}{\sqrt{6}} = 1.96 \cdot 1.224744871391589 = 2.4004558381776553
\][/tex]
3. Construct the 95% Confidence Interval:
The confidence interval is calculated as:
[tex]\[
(\bar{x} - E, \bar{x} + E)
\][/tex]
So,
[tex]\[
(68.16666666666667 - 2.4004558381776553, 68.16666666666667 + 2.4004558381776553)
\][/tex]
Simplifying the bounds:
[tex]\[
(65.76621082848902, 70.56712250484432)
\][/tex]
Thus, the detailed steps yield:
- Sample Mean ([tex]\(\bar{x}\)[/tex]): 68.16666666666667
- Margin of Error at 95% Confidence Level: 2.4004558381776553
- 95% Confidence Interval: (65.76621082848902, 70.56712250484432)
