Question

a score of x = 75 is measured in a population with a mean of μ = 100. a z - score of z = +1.50 is calculated. without knowing the standard deviation, explain why the z - score of z = +1.50 is incorrect.

Explanation:

Step1: Recall 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 standard deviation.

Step2: Analyze given values

We have $X = 75$ and $\mu=100$. Substituting into the formula gives $z=\frac{75 - 100}{\sigma}=\frac{- 25}{\sigma}$. Since $\sigma>0$, the value of $z$ will be negative.

Answer:

The raw score $X = 75$ is less than the mean $\mu = 100$. By the z - score formula, when $X<\mu$, $z=\frac{X - \mu}{\sigma}<0$. So a positive z - score of $z = + 1.50$ is incorrect for $X = 75$ and $\mu = 100$.