7.18 Vail Resorts pays part-time seasonal employees at ski resorts on an hourly basis. At a certain mountain, the hourly rates have a normal distribution with $\sigma =$ \$3.00. If 20 percent of all part- time seasonal employees make more than \$13.16 an hour, what is the average hourly pay rate at this mountain?

#### Solution

Let $X$ denote the hourly pay rates. $X\sim N(\mu,3^2)$.

Let $\mu$ be the average hourly pay rate at this mountain.

` $$ \begin{aligned} &P(X\geq 13.16) =0.2\\ &\Rightarrow P(X < 13.16) =0.8\\ &\Rightarrow P\big(\frac{X-\mu}{\sigma} < \frac{13.16-\mu}{3}\big)=0.8\\ &\Rightarrow P(Z < \frac{13.16-\mu}{3}\big)=0.8\\ &\Rightarrow \frac{13.16-\mu}{3}= 0.842\\ & \quad \quad \text{(from normal table)}\\ &\Rightarrow \mu = 13.16 - 0.842* 3\\ &\Rightarrow \mu = 10.634 \end{aligned} $$ `

The average hourly pay rate at this mountain is $\mu=$ \$ $10.634$ .

#### Further Reading

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