Let's solve the problem step-by-step:
### Step 1: Understanding the Problem
You are given the mean ([tex]\(\mu\)[/tex]), standard deviation ([tex]\(\sigma\)[/tex]), and a [tex]\(z\)[/tex]-score. You need to determine which of the given credit scores fall within a [tex]\(z\)[/tex]-score of 3.3 from the mean.
### Step 2: Calculate the Boundaries
First, we will calculate the lower and upper boundaries for the [tex]\(z\)[/tex]-score of 3.3:
- Mean ([tex]\(\mu\)[/tex]): 690
- Standard Deviation ([tex]\(\sigma\)[/tex]): 14
- [tex]\(z\)[/tex]-score: 3.3
The formula for finding the boundary values with a [tex]\(z\)[/tex]-score is:
[tex]\[ \text{Boundary} = \mu \pm (z \times \sigma) \][/tex]
So, we calculate:
#### Lower Boundary:
[tex]\[ \text{Lower Bound} = \mu - (z \times \sigma) = 690 - (3.3 \times 14) \][/tex]
[tex]\[ \text{Lower Bound} = 690 - 46.2 \][/tex]
[tex]\[ \text{Lower Bound} = 643.8 \][/tex]
#### Upper Boundary:
[tex]\[ \text{Upper Bound} = \mu + (z \times \sigma) = 690 + (3.3 \times 14) \][/tex]
[tex]\[ \text{Upper Bound} = 690 + 46.2 \][/tex]
[tex]\[ \text{Upper Bound} = 736.2 \][/tex]
### Step 3: Determine the Scores within the Boundaries
Now we have the boundaries:
- Lower Boundary: 643.8
- Upper Boundary: 736.2
We need to check which of the given credit scores fall within this range:
- 634
- 640
- 720
- 750
#### Checking Each Score:
- 634 is not within 643.8 and 736.2.
- 640 is not within 643.8 and 736.2.
- 720 is within 643.8 and 736.2.
- 750 is not within 643.8 and 736.2.
### Step 4: Conclusion
Among the given choices, the credit score of 720 is the only one that falls within a [tex]\(z\)[/tex]-score of 3.3 from the mean.
Thus, the credit score within a [tex]\(z\)[/tex]-score of 3.3 is:
[tex]\[ \boxed{720} \][/tex]
