Let's break down how to find the mean ([tex]\(\mu\)[/tex]) and the standard deviation ([tex]\(\sigma\)[/tex]) of the normal distribution where the middle 10% of the area lies between 7 and 13:
### 1. Understanding the Problem
We are given an interval [tex]\((7, 13)\)[/tex] where the middle 10% of the area under the normal distribution curve lies. This implies [tex]\(7 \leq x \leq 13\)[/tex] encompasses the middle 10% of the data. Hence, 10% of the area under the curve lies within this interval.
### 2. Given Information and Symbols
- Middle percentage area: 10% ([tex]\(0.10\)[/tex])
- Lower bound of the interval: [tex]\(7\)[/tex]
- Upper bound of the interval: [tex]\(13\)[/tex]
- Mean ([tex]\(\mu\)[/tex]): To be determined
- Standard deviation ([tex]\(\sigma\)[/tex]): To be determined
### 3. Finding the Mean ([tex]\(\mu\)[/tex])
We already received information that the mean, [tex]\(\mu\)[/tex], is 10. This is derived based on the symmetry of the normal distribution and the interval provided.
[tex]\[
\boxed{\mu = 10}
\][/tex]
### 4. Finding the Standard Deviation ([tex]\(\sigma\)[/tex])
To find [tex]\(\sigma\)[/tex], we need to calculate the z-scores for the lower and upper bounds of the interval and match them with their respective cumulative probabilities under the standard normal curve.
Given the final result from the computations:
- Lower z-score: [tex]\(-1.6448270190251657\)[/tex]
- Upper z-score: [tex]\(1.6448270190251657\)[/tex]
Using the z-score formula:
[tex]\[
z = \frac{(X - \mu)}{\sigma}
\][/tex]
For the lower bound ([tex]\(X = 7\)[/tex]):
[tex]\[
-1.6448270190251657 = \frac{(7 - 10)}{\sigma}
\][/tex]
[tex]\[
\sigma = \frac{(10 - 7)}{1.6448270190251657} \approx 1.8239
\][/tex]
For the upper bound ([tex]\(X = 13\)[/tex]):
[tex]\[
1.6448270190251657 = \frac{(13 - 10)}{\sigma}
\][/tex]
[tex]\[
\sigma = \frac{(13 - 10)}{1.6448270190251657} \approx 1.8239
\][/tex]
Thus, the standard deviation is:
[tex]\[
\boxed{\sigma = 1.8239}
\][/tex]
### Conclusion
The correct answers are:
[tex]\[
\text{a) } \mu = 10
\][/tex]
[tex]\[
\text{b) } \sigma = 1.8239
\][/tex]
If there was any submission error, it should be re-evaluated contextually to ensure the correct interpretation.
