1. **State the problem:** We have a normal distribution with mean $\mu = 20$ and standard deviation $\sigma = 5$. We want to find the probability $P(15 < x < 25)$. 2. **Formula and rules:** For a normal distribution, probabilities are found using the standard normal variable $Z = \frac{X - \mu}{\sigma}$. 3. **Convert bounds to $Z$-scores:** $$Z_1 = \frac{15 - 20}{5} = \frac{-5}{5} = -1$$ $$Z_2 = \frac{25 - 20}{5} = \frac{5}{5} = 1$$ 4. **Find the probability:** $$P(15 < x < 25) = P(-1 < Z < 1)$$ 5. Using standard normal distribution tables or symmetry: $$P(-1 < Z < 1) = P(Z < 1) - P(Z < -1)$$ 6. From the standard normal table: $$P(Z < 1) = 0.8413$$ $$P(Z < -1) = 0.1587$$ 7. Calculate the probability: $$P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826$$ **Final answer:** $$\boxed{0.6826}$$