2eachers' Salaries The average annual salary for all U.S. teachers is \( \$ 47,750 \). Assume that the distribution is normal and the standard deviation is \( \$ 5680 \). Find the probability that a randomly selected teacher earns a. Between \( \$ 35,000 \) and \( \$ 45,000 \) a year b. More than \( \$ 40,000 \) a year c. If you were applying for a teaching position and were offered \( \$ 31,000 \) a year, how would you feel (based on this information)?

To analyze the salaries of U.S. teachers based on the given normal distribution parameters, we'll use the properties of the **normal distribution**. Here's a detailed breakdown for each part of your question: - **Mean (μ)**: \$47,750 - **Standard Deviation (σ)**: \$5,680 ### a. Probability of a Teacher Earning Between \$35,000 and \$45,000 a Year We want to find \( P(35,000 \leq X \leq 45,000) \). 1. **Convert the salaries to z-scores:** \[ z_1 = \frac{35,000 - 47,750}{5,680} \approx -2.2458 \] \[ z_2 = \frac{45,000 - 47,750}{5,680} \approx -0.4841 \] 2. **Find the corresponding probabilities using the standard normal distribution table:** - \( P(Z \leq -2.2458) \approx 0.0125 \) - \( P(Z \leq -0.4841) \approx 0.3142 \) 3. **Calculate the probability between these z-scores:** \[ P(-2.2458 \leq Z \leq -0.4841) = 0.3142 - 0.0125 = 0.3017 \] **So, the probability is approximately **30.2%**.** ### b. Probability of a Teacher Earning More Than \$40,000 a Year We want to find \( P(X > 40,000) \). 1. **Convert the salary to a z-score:** \[ z = \frac{40,000 - 47,750}{5,680} \approx -1.3662 \] 2. **Find the corresponding probability:** Since the normal distribution is symmetric, \[ P(Z > -1.3662) = P(Z < 1.3662) \approx 0.9141 \] **So, the probability is approximately **91.4%**.** ### c. Interpretation of Being Offered \$31,000 a Year To understand how unusual an offer of \$31,000 is, let's determine its position relative to the average. 1. **Convert \$31,000 to a z-score:** \[ z = \frac{31,000 - 47,750}{5,680} \approx -2.9472 \] 2. **Find the corresponding probability:** \[ P(Z < -2.9472) \approx 0.0016 \text{ or } 0.16\% \] This extremely low probability indicates that earning \$31,000 is highly uncommon among U.S. teachers. **Conclusion for Part c:** An offer of **\$31,000** is **significantly below the average teacher salary**. Statistically, such a salary is in the **bottom 0.2%** of earners, suggesting that the offer is unusually low compared to typical salaries. If you were offered this amount, it would be advisable to **negotiate** or **seek further information** about the role, benefits, and any other compensations that might offset the lower salary.