The average salary of a male full professor at a public four-year institution offering classes at the doctoral level is \$109628. For a female full professor at the same kind of institution, the salary is \$79330. If the standard deviation for the salaries of both genders is approximately \$5200 and the salaries are normally distributed, find the 80th percentile salary for male professors and for female professors.

### Solution

Given that the average salary of a male full professor at a public four-year institution offering classes at the doctoral level is $\mu_m= 109628$.

A female full professor at the same kind of institution, the average salary is $\mu_f =79330$.

The standard deviation for the salaries of both genders is approximately $\sigma = 5200$.

- Let the 80th percentile for male professor be "a". Then we have

` $$ \begin{aligned} & P(X\leq a) =0.8\\ &\Rightarrow P\big(\frac{X-\mu_m}{\sigma} < \frac{a-109628}{5200}\big)=0.8\\ &\Rightarrow P(Z < \frac{a-109628}{5200}\big)=0.8\\ &\Rightarrow \frac{a-109628}{5200}= 0.842\\ &\Rightarrow a = 109628 + 0.842* 5200\\ &\Rightarrow a = 114006.4 \end{aligned} $$ `

Thus the 80th percentile salary for male professor is \$ $114006.4$.

- Let the 80th percentile salary for Female professors be "b". Then we have

` $$ \begin{aligned} & P(X\leq b) =0.8\\ &\Rightarrow P\big(\frac{X-\mu_f}{\sigma} < \frac{b-79330}{5200}\big)=0.8\\ &\Rightarrow P(Z < \frac{b-79330}{5200}\big)=0.8\\ &\Rightarrow \frac{b-79330}{5200}= 0.842\\ &\Rightarrow b = 79330 + 0.842* 5200\\ &\Rightarrow b = 83708.4 \end{aligned} $$ `

Thus the 80th percentile salary for female professor is \$ $83708.4$.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators