1. **Problem Statement:** We have a population of student heights normally distributed with mean $\mu=168$ cm and standard deviation $\sigma=5.0$ cm. We need to find z-scores for heights 175 cm and 164 cm, find the percentage of students between these heights, and find the percentage taller than 175 cm. 2. **Formula for z-score:** $$z=\frac{X-\mu}{\sigma}$$ where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. 3. **Calculate z-score for 175 cm:** $$z=\frac{175-168}{5}=\frac{7}{5}=1.4$$ 4. **Calculate z-score for 164 cm:** $$z=\frac{164-168}{5}=\frac{-4}{5}=-0.8$$ 5. **Find percentage between 164 cm and 175 cm:** Look up the cumulative probabilities for $z=1.4$ and $z=-0.8$ in the standard normal table or use a calculator. - $P(Z<1.4) \approx 0.9192$ - $P(Z<-0.8) \approx 0.2119$ Percentage between = $0.9192 - 0.2119 = 0.7073$ or 70.73% 6. **Find percentage for heights 175 cm or more:** This is $P(Z \geq 1.4) = 1 - P(Z < 1.4) = 1 - 0.9192 = 0.0808$ or 8.08% **Final answers:** - a) $z=1.4$ - b) $z=-0.8$ - c) Approximately 70.73% of students have heights between 164 cm and 175 cm. - d) Approximately 8.08% of students have heights 175 cm or more.