1. **Problem statement:** We want to find the lower and upper systolic blood pressure readings that include the middle 75% of the population, assuming a normal distribution with mean $\mu = 120$ and standard deviation $\sigma$. 2. **Understanding the problem:** The middle 75% means the central area under the normal curve between two values $L$ and $U$ such that $P(L < X < U) = 0.75$. This leaves 25% in the tails, split equally as 12.5% in each tail. 3. **Using the standard normal distribution:** We convert to the standard normal variable $Z = \frac{X - \mu}{\sigma}$. We want $P(z_1 < Z < z_2) = 0.75$ where $z_1$ and $z_2$ are the z-scores corresponding to the lower and upper bounds. Because of symmetry, $z_1 = -z$ and $z_2 = z$ for some positive $z$. 4. **Finding the z-score:** The area in each tail is 0.125, so the cumulative probability at $z_1$ is 0.125 and at $z_2$ is 0.875. Using standard normal tables or a calculator, $z \approx 1.15$ satisfies $P(Z < 1.15) = 0.875$. 5. **Converting back to original scale:** Use the formula $$L = \mu + z_1 \sigma = 120 - 1.15 \sigma$$ $$U = \mu + z_2 \sigma = 120 + 1.15 \sigma$$ 6. **Final answer:** The lower and upper blood pressure readings that include the middle 75% of the population are $$\boxed{120 - 1.15 \sigma \text{ and } 120 + 1.15 \sigma}$$ respectively.