1. **State the problem:** We are given the heights of adult men and women in America, both normally distributed with given means and standard deviations. We need to find the z-scores for a man who is 6 feet 3 inches tall and a woman who is 5 feet 11 inches tall, then determine who is relatively taller. 2. **Convert heights to inches:** - Man: 6 feet 3 inches = $6 \times 12 + 3 = 75$ inches - Woman: 5 feet 11 inches = $5 \times 12 + 11 = 71$ inches 3. **Recall the z-score formula:** $$z = \frac{X - \mu}{\sigma}$$ where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. 4. **Calculate the man's z-score:** $$z_{man} = \frac{75 - 69.7}{2.63} = \frac{5.3}{2.63}$$ Intermediate step showing cancellation: $$z_{man} = \frac{\cancel{5.3}}{\cancel{2.63}} = 2.015$$ Rounded to two decimal places: $$z_{man} = 2.02$$ 5. **Calculate the woman's z-score:** $$z_{woman} = \frac{71 - 64.5}{2.57} = \frac{6.5}{2.57}$$ Intermediate step showing cancellation: $$z_{woman} = \frac{\cancel{6.5}}{\cancel{2.57}} = 2.53$$ Rounded to two decimal places: $$z_{woman} = 2.53$$ 6. **Compare z-scores:** The woman has a higher z-score ($2.53$) than the man ($2.02$), meaning she is relatively taller compared to her gender's average height. **Final answers:** - a) Man's z-score: $2.02$ - b) Woman's z-score: $2.53$ - c) The 5 foot 11 inch American woman is relatively taller.