1. **Problem statement:** We have a normally distributed variable $X$ representing the number of days per year with sulfur dioxide emissions above the legal limit. The mean is $\mu=25$ days and the standard deviation is $\sigma=5$ days. 2. **Calculate the z-scores:** - For 15 days: $$z_1=\frac{15-25}{5}=-2$$ - For 35 days: $$z_2=\frac{35-25}{5}=2$$ - For 20 days: $$z_3=\frac{20-25}{5}=-1$$ 3. **Use the standard normal distribution table or calculator:** - $P(15 < X < 35) = P(-2 < Z < 2)$ - $P(X > 35) = P(Z > 2)$ - $P(X < 20) = P(Z < -1)$ 4. **Find probabilities:** - $P(-2 < Z < 2) = \Phi(2) - \Phi(-2) = 0.9772 - 0.0228 = 0.9544$ or 95.44% - $P(Z > 2) = 1 - \Phi(2) = 1 - 0.9772 = 0.0228$ or 2.28% - $P(Z < -1) = \Phi(-1) = 0.1587$ or 15.87% 5. **Interpretation for compliance strategy:** - The company can expect about 95.44% of years to have between 15 and 35 days exceeding limits, indicating this range is typical. - Only about 2.28% of years exceed 35 days, which are high-risk years needing special attention. - About 15.87% of years have fewer than 20 days, showing some variability below average. - These statistics help the company plan for typical and extreme scenarios, allocate resources for mitigation, and manage regulatory risks effectively.