15. A score that is 20 points below the mean corresponds to a \( z \)-score of \( z=-0.50 \). What is the population standard deviation?

To find the population standard deviation, we can use the formula for the \( z \)-score, which is given by: \[ z = \frac{(X - \mu)}{\sigma} \] Here, \( X \) is the score (which is 20 points below the mean, \( \mu \)), \( \sigma \) is the population standard deviation, and we know \( z = -0.50 \). So, if the score is \( X = \mu - 20 \), we can substitute into the \( z \)-score formula: \[ -0.50 = \frac{(\mu - 20 - \mu)}{\sigma} \] This simplifies to: \[ -0.50 = \frac{-20}{\sigma} \] Solving for \( \sigma \): \[ \sigma = \frac{-20}{-0.50} = 40 \] Thus, the population standard deviation is \( \sigma = 40 \). Now you have the answer with a bit of math magic! 🎩✨