To solve this question about the z-score for a credit score of 694 given that the mean ([tex]$\mu$[/tex]) is 690 and the standard deviation ([tex]$\sigma$[/tex]) is 14, follow these steps:
1. Calculate the z-score:
The z-score formula is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
Where [tex]\( X \)[/tex] is the value you're analyzing.
2. Substitute the given values:
[tex]\[
z_{694} = \frac{694 - 690}{14}
\][/tex]
3. Perform the calculation:
[tex]\[
z_{694} = \frac{4}{14} \approx 0.2857142857142857
\][/tex]
4. Interpret the z-score:
- Within 1 Standard Deviation: This means [tex]\( -1 \leq z \leq 1 \)[/tex].
- Between 1 and 2 Standard Deviations: This means [tex]\( 1 < |z| \leq 2 \)[/tex].
- Between 2 and 3 Standard Deviations: This means [tex]\( 2 < |z| \leq 3 \)[/tex].
- More than 3 Standard Deviations: This means [tex]\( |z| > 3 \)[/tex].
Given our computed z-score of approximately 0.286:
- The z-score of 0.286 is within the range [tex]\(-1 \leq z \leq 1\)[/tex]. Therefore, it is within 1 standard deviation of the mean.
- It does not fall in the range [tex]\( 1 < |z| \leq 2 \)[/tex], so it is not between 1 and 2 standard deviations of the mean.
- It does not fall in the range [tex]\( 2 < |z| \leq 3 \)[/tex], so it is not between 2 and 3 standard deviations of the mean.
- It does not fall in the range [tex]\( |z| > 3 \)[/tex], so it is not more than 3 standard deviations of the mean.
Hence, the statement that must be true about [tex]\( z_{694} \)[/tex] is:
[tex]\[ z_{694} \text{ is within 1 standard deviation of the mean} \][/tex]
