1. **Problem Statement:** Given a normal distribution $X \sim N(20, 2.5)$, calculate the following probabilities: 2. **Formula:** Use the standardization formula for normal distribution: $$z = \frac{X - \mu}{\sigma}$$ where $\mu = 20$ and $\sigma = 2.5$. 3. **Step 1: Calculate $P(X > 16)$** - Compute $z$-score: $$z = \frac{16 - 20}{2.5} = -1.6$$ - From standard normal tables, $P(Z < 1.6) = 0.9452$ - Since $P(Z < -1.6) = 1 - P(Z < 1.6) = 1 - 0.9452 = 0.0548$ - Therefore, $$P(X > 16) = 1 - P(X < 16) = 1 - 0.0548 = 0.9452$$ 4. **Step 2: Calculate $P(X < 21)$** - Compute $z$-score: $$z = \frac{21 - 20}{2.5} = 0.4$$ - From standard normal tables, $P(Z < 0.4) = 0.6554$ - So, $$P(X < 21) = 0.6554$$ 5. **Step 3: Calculate $P(15 < X < 22)$** - Compute $z$-scores: - $$z_{15} = \frac{15 - 20}{2.5} = -2.0$$ - $$z_{22} = \frac{22 - 20}{2.5} = 0.8$$ - From tables: - $P(Z < 2.0) = 0.9772$ so $P(Z < -2.0) = 1 - 0.9772 = 0.0228$ - $P(Z < 0.8) = 0.7881$ - Therefore, $$P(15 < X < 22) = P(Z < 0.8) - P(Z < -2.0) = 0.7881 - 0.0228 = 0.7653$$ 6. **Step 4: Calculate $P(14 < X < 18)$** - Compute $z$-scores: - $$z_{14} = \frac{14 - 20}{2.5} = -2.4$$ - $$z_{18} = \frac{18 - 20}{2.5} = -0.8$$ - From tables: - $P(Z < 2.4) = 0.9918$ so $P(Z < -2.4) = 1 - 0.9918 = 0.0082$ - $P(Z < 0.8) = 0.7881$ so $P(Z < -0.8) = 1 - 0.7881 = 0.2119$ - Therefore, $$P(14 < X < 18) = P(Z < -0.8) - P(Z < -2.4) = 0.2119 - 0.0082 = 0.2037$$ **Conclusion:** All calculations and logic are correct based on the standard normal distribution and the given values.