Sure, let's go through this step-by-step to determine the mean height, margin of error, and the confidence interval at a 90% confidence level.
### 1. Calculate the sample mean ( [tex]\(\bar{x}\)[/tex] ):
The sample data given is:
[tex]\[71, 68, 73, 60, 62, 74, 68, 68, 61, 60\][/tex]
To find the sample mean, you add up all the heights and divide by the number of students (sample size):
[tex]\[
\bar{x} = \frac{71 + 68 + 73 + 60 + 62 + 74 + 68 + 68 + 61 + 60}{10} = \frac{665}{10} = 66.5
\][/tex]
### 2. Calculate the margin of error at a 90% confidence level:
The margin of error (ME) can be calculated using the formula:
[tex]\[
ME = Z_{\alpha/2} \times \left( \frac{\sigma}{\sqrt{n}} \right)
\][/tex]
where:
- [tex]\( Z_{\alpha/2} \)[/tex] is the z-score corresponding to the desired confidence level.
- [tex]\( \sigma \)[/tex] is the population standard deviation.
- [tex]\( n \)[/tex] is the sample size.
Given:
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 2 inches
- Sample size ([tex]\(n\)[/tex]) = 10
- Confidence level = 90%
For a 90% confidence level, the z-score ([tex]\( Z_{\alpha/2} \)[/tex]) is approximately 1.645.
Now, plug these values into the margin of error formula:
[tex]\[
ME = 1.645 \times \left( \frac{2}{\sqrt{10}} \right) = 1.645 \times 0.6325 = 1.0402967757511148
\][/tex]
### 3. Calculate the 90% confidence interval:
The confidence interval is calculated as:
[tex]\[
\left( \bar{x} - ME, \bar{x} + ME \right)
\][/tex]
Using the sample mean ([tex]\(\bar{x}\)[/tex]) and the margin of error (ME) we calculated:
[tex]\[
\text{Confidence interval} = (66.5 - 1.0402967757511148, 66.5 + 1.0402967757511148) = (65.45970322424888, 67.54029677575112)
\][/tex]
### Summary
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 66.5
- Margin of error at 90% confidence level = 1.0402967757511148
- 90% confidence interval = [tex]\([65.45970322424888, 67.54029677575112]\)[/tex]
Thus, the detailed calculations provide:
[tex]\[
\bar{x} = 66.5
\][/tex]
Margin of error at [tex]\(90\% \)[/tex] confidence level [tex]\(= 1.0402967757511148 \)[/tex]
[tex]\[ 90 \% \text{ confidence interval } = [65.45970322424888, 67.54029677575112] \][/tex]
