Question

In a distribution, a score of X=60 has a z-score of -3.0.. A score of X=85 has a z-score of 2.0. What are the mean and standard deviation for this population?

Answer 1

The formula for a z-score is
$z=\frac{X- \mu}{\sigma}$
We don't have μ, the mean, or σ, the standard deviation.  We can, however, set up a system of two equations with the information we do have.  We know:
$-3=\frac{60- \mu}{\sigma} \text{ and } 2=\frac{85- \mu}{\sigma}$
We can make this look more like a "typical" system of equations by cancelling the fractional part of each equation.  We do this by multiplying the bottom of each equation by σ:
$-3*\sigma=\frac{60- \mu}{\sigma}*\sigma \text{ and } 2*\sigma=\frac{85- \mu}{\sigma}*\sigma \\ \\-3\sigma=60- \mu \text{ and } 2\sigma=85- \mu$
Now we'll get the variables on the same side of the equation.  We'll do this by adding μ to both sides:
$-3\sigma + \mu=60- \mu + \mu \text{ and } 2\sigma+ \mu =85- \mu + \mu \\ \\-3\sigma+\mu=60 \text{ and } 2\sigma+\mu=85$
Now we'll write this vertically:
$\left \{ {{-3\sigma+\mu=60} \atop {2\sigma+\mu=85}} \right.$
We will "eliminate" μ by subtracting the bottom equation:
$\left \{ {{-3\sigma+\mu=60} \atop {-(2\sigma+\mu=85)}} \right. \\ \\-5\sigma=-25$
Divide both sides by -5:
$\frac{-5\sigma}{-5}=\frac{-25}{-5} \\ \\\sigma=5$
Our standard deviation is 5.  Substituting this into our top equation we have:
-3(5)+μ=60
-15+μ=60
Cancel the -15 by adding it to both sides:
-15+μ+15=60+15
μ=75
Our mean is 75.