Question

sample standard deviation? 22. 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 population mean, and $\sigma$ is the population standard deviation.
For the first score $X_1 = 56$ with $z_1=-0.40$, we have $- 0.40=\frac{56-\mu}{\sigma}$, which can be rewritten as $-0.40\sigma=56 - \mu$ or $\mu=56 + 0.40\sigma$.
For the second score $X_2 = 70$ with $z_2 = 1.00$, we have $1.00=\frac{70-\mu}{\sigma}$, which can be rewritten as $\sigma=70 - \mu$.

Step2: Substitute $\mu$ from the first - derived equation into the second - derived equation

Substitute $\mu = 56+0.40\sigma$ into $\sigma=70 - \mu$.
We get $\sigma=70-(56 + 0.40\sigma)$.
Expand the right - hand side: $\sigma=70 - 56-0.40\sigma$.
Combine like terms: $\sigma+0.40\sigma=14$.
$1.40\sigma=14$.
Solve for $\sigma$: $\sigma=\frac{14}{1.40}=10$.

Step3: Find the mean $\mu$

Substitute $\sigma = 10$ into $\mu=56 + 0.40\sigma$.
$\mu=56+0.40\times10$.
$\mu=56 + 4=60$.

Answer:

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