Sure! Let's work through this problem step-by-step to find the z-score for a value of 74 in a normal distribution with a mean of 62 and a standard deviation of 12.
1. Identify the given values:
- Mean (μ) = 62
- Standard Deviation (σ) = 12
- Value (X) = 74
2. Recall the formula for the z-score:
The z-score formula is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where:
- [tex]\(X\)[/tex] is the value for which we want to find the z-score
- [tex]\(\mu\)[/tex] is the mean of the distribution
- [tex]\(\sigma\)[/tex] is the standard deviation of the distribution
3. Substitute the given values into the formula:
[tex]\[
z = \frac{74 - 62}{12}
\][/tex]
4. Calculate the difference in the numerator:
[tex]\[
74 - 62 = 12
\][/tex]
5. Divide the result by the standard deviation:
[tex]\[
z = \frac{12}{12} = 1
\][/tex]
Therefore, the z-score for a value of 74 in a normal distribution with a mean of 62 and a standard deviation of 12 is [tex]\(\boxed{1}\)[/tex].
