1. **State the problem:** Find the area under the normal distribution curve between $70 < X < 95$ where the mean $\overline{x} = 100$ and standard deviation $\sigma = 10$. 2. **Formula and rules:** We use the standard normal distribution formula and convert $X$ values to $Z$ scores using: $$Z = \frac{X - \overline{x}}{\sigma}$$ The area between two $X$ values corresponds to the probability between their $Z$ scores. 3. **Calculate $Z$ scores:** $$Z_1 = \frac{70 - 100}{10} = \frac{-30}{10} = -3$$ $$Z_2 = \frac{95 - 100}{10} = \frac{-5}{10} = -0.5$$ 4. **Find area between $Z_1$ and $Z_2$:** Using standard normal distribution tables or a calculator, $$P(Z < -0.5) \approx 0.3085$$ $$P(Z < -3) \approx 0.0013$$ 5. **Calculate the area between:** $$P(-3 < Z < -0.5) = P(Z < -0.5) - P(Z < -3) = 0.3085 - 0.0013 = 0.3072$$ 6. **Interpretation:** The area under the curve between $X=70$ and $X=95$ is approximately $0.3072$, or 30.72%.