Sure! Let's solve this step by step:
1. Understand the given information:
- The mean (average) credit score, [tex]\(\mu\)[/tex], is 690.
- The standard deviation, [tex]\(\sigma\)[/tex], is 14.
- We are looking for the credit score that corresponds to a [tex]\(z\)[/tex]-score of 3.3.
2. Calculate the corresponding credit score for a [tex]\(z\)[/tex]-score of 3.3:
The formula for converting a [tex]\(z\)[/tex]-score to an actual score in the context of normal distribution is:
[tex]\[
X = \mu + z \times \sigma
\][/tex]
where:
- [tex]\(X\)[/tex] is the credit score we are trying to find,
- [tex]\(\mu\)[/tex] is the mean,
- [tex]\(z\)[/tex] is the [tex]\(z\)[/tex]-score,
- [tex]\(\sigma\)[/tex] is the standard deviation.
Substituting the given values:
[tex]\[
X = 690 + 3.3 \times 14
\][/tex]
3. Perform the multiplication and addition:
[tex]\[
3.3 \times 14 = 46.2
\][/tex]
[tex]\[
X = 690 + 46.2 = 736.2
\][/tex]
4. Compare 736.2 against the given credit scores:
The available credit scores are: 634, 640, 720, and 750. We are looking for the score that is closest to 736.2.
5. Determine the closest credit score to 736.2:
- The difference between 736.2 and 634 is 102.2.
- The difference between 736.2 and 640 is 96.2.
- The difference between 736.2 and 720 is 16.2.
- The difference between 736.2 and 750 is 13.8.
The smallest difference is 13.8, which corresponds to the credit score of 750.
Conclusion:
The credit score within a [tex]\(z\)[/tex]-score of 3.3 closest to 736.2 is [tex]\(\boxed{750}\)[/tex].
