To determine the [tex]$X$[/tex] value corresponding to a [tex]$z$[/tex]-score of 1.50 for a population with a mean ([tex]$\mu$[/tex]) of 40 and a standard deviation ([tex]$\sigma$[/tex]) of 8, we can use the formula for converting a [tex]$z$[/tex]-score to an [tex]$X$[/tex] value in a normal distribution:
[tex]\[ X = \mu + z \cdot \sigma \][/tex]
Here's a step-by-step solution:
1. Identify the given values:
- Population mean ([tex]$\mu$[/tex]): 40
- Population standard deviation ([tex]$\sigma$[/tex]): 8
- [tex]$z$[/tex]-score ([tex]$z$[/tex]): 1.50
2. Substitute the given values into the formula:
[tex]\[ X = 40 + 1.50 \cdot 8 \][/tex]
3. Perform the multiplication:
[tex]\[ 1.50 \cdot 8 = 12 \][/tex]
4. Add the product to the mean:
[tex]\[ X = 40 + 12 = 52 \][/tex]
Therefore, the [tex]$X$[/tex] value corresponding to [tex]$z = 1.50$[/tex] is 52.
The correct answer is:
C. 52
