Question

in a population distribution, a score of x = 56 corresponds to z = -0.40 and a score of x = 70 corresponds to z = +1.00. find the mean and standard deviation for the population. (hint: sketch the distribution and locate the two scores on your sketch.)

Explanation:

Step1: Recall the z - score formula

The z - score formula is $z=\frac{X - \mu}{\sigma}$, where $X$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation.
For $X = 56$ and $z=- 0.40$, we have $-0.40=\frac{56-\mu}{\sigma}$, which can be rewritten as $-0.40\sigma=56 - \mu$ or $\mu-0.40\sigma = 56$.
For $X = 70$ and $z = 1.00$, we have $1.00=\frac{70-\mu}{\sigma}$, which can be rewritten as $\sigma=70 - \mu$ or $\mu+\sigma=70$.

Step2: Solve the system of equations

We have the following system of equations:

$$\begin{cases}\mu-0.40\sigma = 56\\\mu+\sigma=70\end{cases}$$

Subtract the first equation from the second equation:
$(\mu+\sigma)-(\mu - 0.40\sigma)=70 - 56$.
Expand the left - hand side: $\mu+\sigma-\mu + 0.40\sigma=70 - 56$.
Combine like terms: $(1 + 0.40)\sigma=14$, so $1.4\sigma=14$.
Solve for $\sigma$: $\sigma=\frac{14}{1.4}=10$.

Step3: Find the mean

Substitute $\sigma = 10$ into the equation $\mu+\sigma=70$.
We get $\mu+10 = 70$.
Solve for $\mu$: $\mu=70 - 10=60$.

Answer:

The mean $\mu = 60$ and the standard deviation $\sigma = 10$.