Question

1. a population with μ = 90 and σ = 20 is transformed into z - scores. after the transformation, what is the mean for the population of z - scores? a. μ = 80 b. μ = 1.00 c. μ = 0 d. cannot be determined from the information given.

Explanation:

Step1: Recall z - score formula property

The formula for a z - score is $z=\frac{x - \mu}{\sigma}$, where $x$ is a raw score, $\mu$ is the population mean, and $\sigma$ is the population standard deviation. When we transform an entire population of scores into z - scores, the mean of the z - scores is always 0. This is a fundamental property of the standard normal distribution. Mathematically, if we take the mean of the z - scores $\mu_z=\frac{\sum_{i = 1}^{N}\frac{x_i-\mu}{\sigma}}{N}$, and using the properties of the mean $\sum_{i = 1}^{N}(x_i - \mu)=\sum_{i = 1}^{N}x_i-N\mu$, and since $\mu=\frac{\sum_{i = 1}^{N}x_i}{N}$, we have $\sum_{i = 1}^{N}(x_i - \mu)=0$. So $\mu_z = 0$.

Answer:

C. $\mu = 0$