Answer to Question #171053 in Statistics and Probability for Melissa M Elliott

Question

The average salary of a male full professor at a public four-year institution offering classes at the doctoral level is $97,750. For a female full professor at the same kind of institution, the salary is $90,280. If the standard deviation for the salaries of both genders is approximately $4700 and the salaries are normally distributed, find the 80th percentile salary for the male professors and for female professors, respectively.

Answer 1

According to distribution of a normal random variable with mean = m and standard variance = $\sigma$ :

$Pr(X < q) = \Phi(\frac{q - m}{\sigma}) = \alpha$ , where $\Phi(x)$ is distribution function of a normal random variable with mean = 0 and standard deviation = 1, q is $\alpha$ -quantile (percentile)

Then $q = m + \sigma \Phi^{-1}(\alpha)$

For male professors: $q = 97750 + 4700 \Phi^{-1}(0.8) = 101706$

For female professors: $q = 90280 + 4700 \Phi^{-1}(0.8) = 94236$