We are asked to estimate the mean height of all 9th grade students at a high school using the heights of 9 randomly selected students. Given the population standard deviation [tex]\( \sigma = 2 \)[/tex] inches, we can find the sample mean, margin of error at 90% confidence level, and the 90% confidence interval for the population mean.
Here is the detailed, step-by-step process:
### Step 1: Calculate the Sample Mean
First, we need to find the sample mean ([tex]\( \bar{x} \)[/tex]) of the 9 selected students' heights. The given heights are [tex]\( 61, 62, 71, 62, 62, 63, 63, 66, 62 \)[/tex].
[tex]\[
\bar{x} = \frac{61 + 62 + 71 + 62 + 62 + 63 + 63 + 66 + 62}{9} = \frac{512}{9} \approx 63.56
\][/tex]
### Step 2: Calculate the Margin of Error
To calculate the margin of error (ME) at a 90% confidence level, we need the Z-score for a 90% confidence level. The Z-score for a 90% confidence level (which corresponds to 95% cumulative from the left tail) is approximately 1.645.
The formula for the margin of error is:
[tex]\[
\text{Margin of Error} (ME) = Z \times \left( \frac{\sigma}{\sqrt{n}} \right)
\][/tex]
Where [tex]\( Z \)[/tex] is the Z-score (1.645 for 90% confidence), [tex]\( \sigma \)[/tex] is the population standard deviation (2 inches), and [tex]\( n \)[/tex] is the sample size (9).
[tex]\[
ME = 1.645 \times \left( \frac{2}{\sqrt{9}} \right) = 1.645 \times \frac{2}{3} \approx 1.10
\][/tex]
### Step 3: Calculate the 90% Confidence Interval
Finally, we can calculate the 90% confidence interval for the population mean using the formula:
[tex]\[
\text{Confidence Interval} = (\bar{x} - ME, \bar{x} + ME)
\][/tex]
Substituting the values we have:
[tex]\[
\text{Confidence Interval} = (63.56 - 1.10, 63.56 + 1.10) = (62.46, 64.65)
\][/tex]
### Summary
- Sample Mean ([tex]\( \bar{x} \)[/tex]) = 63.56
- Margin of Error at 90% confidence level = 1.10
- 90% Confidence Interval = (62.46, 64.65)
So, the final detailed solution is:
[tex]\[
\begin{array}{l}
\bar{x}=63.56 \\
\text{Margin of error at } 90\% \text{ confidence level} = 1.10 \\
90\% \text{ confidence interval} = \left[62.46, 64.65\right]
\end{array}
\][/tex]
This interval means that we are 90% confident that the true mean height of all 9th grade students at the high school lies between 62.46 inches and 64.65 inches.
