In a population with [tex]\sigma = 8[/tex], a score of [tex]X = 44[/tex] corresponds to a [tex]z[/tex] score of [tex]z = -0.50[/tex]. What is the population mean?

A. [tex]\mu = 36[/tex]
B. [tex]\mu = 40[/tex]
C. [tex]\mu = 48[/tex]
D. [tex]\mu = 52[/tex]



Answer :

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]