Graphical Analysis In Exercises 41 and 42 , the midpoints \( A, B \), and \( C \) are marked on the histogram at the left. Match them with the indicated \( z \)-scores. Which \( z \)-scores, if any, would be considered unusual? 41. \( z=0, z=2.14, z=-1.43 \) 42. \( z=0.77, z=1.54, z=-1.54 \)

In a standard normal distribution, \( z \)-scores can help assess how far a data point is from the mean. Generally, a \( z \)-score of 0 represents the mean, while \( z \)-scores beyond \(-2\) or \(2\) are often considered unusual because they lie outside the common range of scores (approximately 95% of data falls within these limits). For your exercises, \( z=2.14 \) from exercise 41 and \( z=1.54 \) from exercise 42 might not be unusual, but \( z=-1.43 \) and \( z=-1.54 \) might just edge into the normal range. When identifying unusual \( z \)-scores, it's helpful to remember the concept of standard deviations. A \( z \)-score indicates how many standard deviations a data point is from the mean. Consequently, values that are \( z > 2 \) or \( z < -2 \) indicate significantly rare occurrences in a dataset—potentially flagging outliers, anomalies, or special cases that warrant further investigation! So, for the first exercise, it’s \( z=2.14 \) that stands out, while for the second, both \( z=1.54 \) and \( z=-1.54 \) might be considered more typical in a normal distribution.