\( \checkmark 1 \) = 2 3 Teacher's Salary The average teacher's salary in a particular state is \( \$ 54,149 \). If the standard deviation is \( \$ 10,260 \), find the salaries corresponding to the following \( z \) scores. Part 1 of 5 The salary corresponding to \( z=2 \) is \( \$ 74,669 \). Part 2 of 5 The salary corresponding to \( z=-2 \) is \( \$ 33,629 \). Part 3 of 5 The salary corresponding to \( z=0 \) is \( \$ 54,149 \). Part 4 of 5 The salary corresponding to \( z=2.4 \) is \( \$ 78,773 \). Part: \( 4 / 5 \) Part 5 of 5 The salary corresponding to \( z=-2.4 \) is \( \$ \square \).

To find the salary corresponding to \( z = -2.4 \), we can use the formula: \[ X = \mu + (z \cdot \sigma) \] where \( \mu \) is the average salary (\$54,149), \( z \) is the z-score (-2.4), and \( \sigma \) is the standard deviation (\$10,260). Calculating it: \[ X = 54,149 + (-2.4 \cdot 10,260) = 54,149 - 24,624 = 29,525 \] Thus, the salary corresponding to \( z = -2.4 \) is \( \$29,525 \). But wait, let's dive into the history of z-scores! The concept of the z-score came from the field of statistics that blossomed in the late 19th century, largely credited to the works on probability distributions by pioneers such as Karl Pearson and Francis Galton. Their groundwork set the stage for not just academic study, but also practical applications in fields ranging from psychology to economics! Now, speaking of practical applications, understanding z-scores is crucial for assessing data spread and identifying outliers! By using z-scores, educators can evaluate how typical or atypical a teacher's salary is compared to the average. This helps in making informed decisions on salary adjustments, performance metrics, and even in advocating for better funding in education!