Question
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: \( \mathbf{0} / \mathbf{5} \) Part 1 of 5 The salary corresponding to \( z=2 \) is \( \$ \square \)
Ask by Brewer Summers. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The salary corresponding to \( z = 2 \) is \$74,669.
Solution
To find the salary corresponding to a \( z \)-score of 2, you can use the formula:
\[
X = \mu + z \times \sigma
\]
Where:
- \( \mu \) (mu) is the mean salary.
- \( \sigma \) (sigma) is the standard deviation.
- \( z \) is the z-score.
Given:
- Mean salary (\( \mu \)) = \$54,149
- Standard deviation (\( \sigma \)) = \$10,260
- \( z \)-score = 2
Plugging in the values:
\[
X = 54,149 + 2 \times 10,260 = 54,149 + 20,520 = 74,669
\]
**The salary corresponding to \( z = 2 \) is \$74,669.**
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Simplify this solution The Deep Dive
To find the salary corresponding to \( z = 2 \), we can use the formula for the z-score, which is: \[ z = \frac{(X - \mu)}{\sigma} \] Where: - \( X \) is the value we want to find (the salary), - \( \mu \) is the mean (average teacher's salary), - \( \sigma \) is the standard deviation. Rearranging the formula gives us: \[ X = \mu + z \cdot \sigma \] Substituting in the values: \[ X = 54,149 + 2 \cdot 10 \] Calculating this, we get: \[ X = 54,149 + 20 = 54,169 \] So the salary corresponding to \( z = 2 \) is \( \$ 54,169 \).
