To determine the population mean (μ) given the standard deviation (σ), a specific score (X), and the corresponding z-score (z), we can use the z-score formula:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
Given:
- Standard deviation, [tex]\(\sigma = 8\)[/tex]
- Score, [tex]\(X = 44\)[/tex]
- z-score, [tex]\(z = -0.50\)[/tex]
We need to find the population mean, [tex]\(\mu\)[/tex]. Rearranging the z-score formula to solve for [tex]\(\mu\)[/tex], we get:
[tex]\[ \mu = X - z \cdot \sigma \][/tex]
Substitute the given values into the equation:
[tex]\[ \mu = 44 - (-0.50) \cdot 8 \][/tex]
First, calculate the product of the z-score and the standard deviation:
[tex]\[ -0.50 \cdot 8 = -4 \][/tex]
Then, substitute this value back into the equation:
[tex]\[ \mu = 44 - (-4) \][/tex]
Subtracting a negative number is the same as adding the positive equivalent:
[tex]\[ \mu = 44 + 4 = 48 \][/tex]
Therefore, the population mean is:
[tex]\[ \mu = 48 \][/tex]
So the correct answer is:
C. [tex]\(\mu = 48\)[/tex]
