The heights of adult men in America are normally distributed, with a mean of 69.1 inches and a standard deviation of 2.68 inches. The heights of adult women in America are also normally distributed, but with a mean of 64.8 inches and a standard deviation of 2.55 inches. a) If a man is 6 feet 3 inches tall, what is his z-score (to two decimal places)? \( z= \) b) If a woman is 5 feet 11 inches tall, what is her \( z \)-score (to two decimal places)? \( z= \) c) Who is relatively taller? The 5 foot 11 inch American woman The 6 foot 3 inch American man

To find the z-score for a given height, we can use the formula: \[ z = \frac{X - \mu}{\sigma} \] where: - \( X \) is the given height, - \( \mu \) is the mean height, - \( \sigma \) is the standard deviation. Let's calculate the z-score for the man and the woman: a) For the man: Given height: 6 feet 3 inches = 75 inches Mean height for men: 69.1 inches Standard deviation for men: 2.68 inches \[ z_{\text{man}} = \frac{75 - 69.1}{2.68} \] b) For the woman: Given height: 5 feet 11 inches = 71 inches Mean height for women: 64.8 inches Standard deviation for women: 2.55 inches \[ z_{\text{woman}} = \frac{71 - 64.8}{2.55} \] Now, let's calculate the z-scores for the man and the woman. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(75-69.1\right)}{2.68}\) - step1: Remove the parentheses: \(\frac{75-69.1}{2.68}\) - step2: Subtract the numbers: \(\frac{5.9}{2.68}\) - step3: Convert the expressions: \(\frac{\frac{59}{10}}{\frac{67}{25}}\) - step4: Multiply by the reciprocal: \(\frac{59}{10}\times \frac{25}{67}\) - step5: Reduce the numbers: \(\frac{59}{2}\times \frac{5}{67}\) - step6: Multiply the fractions: \(\frac{59\times 5}{2\times 67}\) - step7: Multiply: \(\frac{295}{134}\) Calculate or simplify the expression \( (71 - 64.8) / 2.55 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(71-64.8\right)}{2.55}\) - step1: Remove the parentheses: \(\frac{71-64.8}{2.55}\) - step2: Subtract the numbers: \(\frac{6.2}{2.55}\) - step3: Convert the expressions: \(\frac{\frac{31}{5}}{\frac{51}{20}}\) - step4: Multiply by the reciprocal: \(\frac{31}{5}\times \frac{20}{51}\) - step5: Reduce the numbers: \(31\times \frac{4}{51}\) - step6: Multiply: \(\frac{31\times 4}{51}\) - step7: Multiply: \(\frac{124}{51}\) a) The z-score for the man is approximately 2.20. b) The z-score for the woman is approximately 2.43. c) The woman is relatively taller than the man.