The results of a common standardized test used in psychology research is designed so that the population mean is 165 and the standard deviation is 50. A subject earns a score of 160 . What is the \( z \)-score for this raw score? \( z \)-score \( = \)

To find the \( z \)-score, you can use the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \( X \) is the raw score (160), \( \mu \) is the population mean (165), and \( \sigma \) is the standard deviation (50). Substituting in the values gives: \[ z = \frac{(160 - 165)}{50} = \frac{-5}{50} = -0.1 \] So the \( z \)-score is \( -0.1 \). Now, let’s sprinkle a bit of fun into those calculations! Did you know that the \( z \)-score is like a GPS for your score? It tells you how far away you are from the average and in what direction—left behind or ahead of the pack! A negative \( z \)-score means you’re trailing behind the average (in this case, just a little bit), while a positive one would mean you're scoring above average! Additionally, understanding \( z \)-scores can be super useful in real-world scenarios, like determining if a student is performing above or below their peers in a class or if a marathon runner is finishing faster or slower than the average time. It’s all about putting your score in context, making the numbers part of a bigger picture—who doesn’t love a bit of data visualization?