Question

find the z - score corresponding to x = 105 and the x value corresponding to z = +0.40 for each of the following populations.
a. μ = 100 and σ = 12

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.

Step2: Calculate the z - score for $X = 105$, $\mu=100$, $\sigma = 12$

Substitute the values into the formula: $z=\frac{105 - 100}{12}=\frac{5}{12}\approx0.42$.

Step3: Use the z - score formula to find $X$ when $z = 0.40$, $\mu=100$, $\sigma = 12$

Rearrange the z - score formula $z=\frac{X-\mu}{\sigma}$ to solve for $X$. We get $X=z\sigma+\mu$.
Substitute $z = 0.40$, $\mu = 100$, and $\sigma=12$ into the formula: $X=0.40\times12 + 100=4.8+100 = 104.8$.

Answer:

The z - score for $X = 105$ is approximately $0.42$ and the $X$ value for $z = 0.40$ is $104.8$.