1. **Problem statement:** We are given the mean height $\mu = 158$ cm, standard deviation $\sigma = 20$ cm, and total number of students $N = 100$. We want to find how many students have heights between 150 cm and 170 cm. 2. **Formula and concept:** We use the properties of the normal distribution. The number of students between two heights corresponds to the probability that a value lies between those heights multiplied by the total number of students. 3. **Calculate z-scores:** The z-score formula is $$z = \frac{x - \mu}{\sigma}$$ - For 150 cm: $$z_1 = \frac{150 - 158}{20} = \frac{-8}{20} = -0.4$$ - For 170 cm: $$z_2 = \frac{170 - 158}{20} = \frac{12}{20} = 0.6$$ 4. **Find probabilities from z-table:** - $P(Z < -0.4) \approx 0.3446$ - $P(Z < 0.6) \approx 0.7257$ 5. **Calculate probability between 150 and 170:** $$P(150 < X < 170) = P(Z < 0.6) - P(Z < -0.4) = 0.7257 - 0.3446 = 0.3811$$ 6. **Find number of students:** $$\text{Number} = 0.3811 \times 100 = 38.11 \approx 38$$ **Final answer:** Approximately 38 students have heights between 150 cm and 170 cm.