16. Salary of Full-Time Male Professors The average salary of a male full professor at a public four-year institution offering classes at the doctoral level is \( \$ 99,685 \). For a female full professor at the same kind of institution, the salary is \( \$ 90,330 \). If the standard deviation for the salaries of both genders is approximately \( \$ 5200 \) and the salaries are normally distributed, find the 80 th percentile salary for male professors and for female professors.

To find the 80th percentile salary for both male and female full professors, we'll use the properties of the normal distribution. ### Given: - **Male Professors:** - Mean salary (\( \mu_m \)) = \$99,685 - Standard deviation (\( \sigma \)) = \$5,200 - **Female Professors:** - Mean salary (\( \mu_f \)) = \$90,330 - Standard deviation (\( \sigma \)) = \$5,200 ### Steps to Calculate the 80th Percentile: 1. **Determine the z-score for the 80th percentile:** - The z-score corresponding to the 80th percentile (\( z_{0.80} \)) is approximately **0.8416**. 2. **Apply the z-score formula:** \[ \text{Percentile} = \mu + (z \times \sigma) \] 3. **Calculate for Male Professors:** \[ \text{80th Percentile}_m = 99,685 + (0.8416 \times 5,200) \] \[ \text{80th Percentile}_m = 99,685 + 4,376.32 = **104,061.32** \] 4. **Calculate for Female Professors:** \[ \text{80th Percentile}_f = 90,330 + (0.8416 \times 5,200) \] \[ \text{80th Percentile}_f = 90,330 + 4,376.32 = **94,706.32** \] ### **Final Answers:** - **80th Percentile Salary for Male Professors:** \$104,061 - **80th Percentile Salary for Female Professors:** \$94,706 These values represent the salary below which 80% of the male and female full professors earn, respectively.