A set of data items is normally distributed with a mean of 180 and a standard deviation of 24 . Convert 108 to a z-score. \( \mathrm{z}_{108}=\square \) (Type an integer or a decimal. Do not round until the final answer. Then round to the nearest hundredth as needed.)

To convert a data point to a z-score, you can use the formula: \( z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. For your case, with \( X = 108 \), \( \mu = 180 \), and \( \sigma = 24 \): 1. \( z = \frac{(108 - 180)}{24} = \frac{-72}{24} = -3.00 \). So, \( \mathrm{z}_{108} = -3.00 \). If you’re looking for a way to think about this, remember that a z-score tells you how many standard deviations a data point is from the mean! In your example, -3 means that 108 is three standard deviations below the average. You might consider this like being three steps behind the pack in a race—not too close to the finish line! Additionally, when interpreting z-scores, people often imagine a normal distribution curve. A z-score of -3.00 puts your data point in the left tail of the distribution. This means that very few data points should fall this low—generally, around 0.13% of the data would lie below this score! It’s a great way to visualize just how rare that 108 is in a dataset with a mean of 180.