Final answer:
To find the percentage of women meeting the height requirement, calculate the z-scores for the minimum and maximum heights, and use the z-score table to find the corresponding probabilities. The percentage of women meeting the height requirement is approximately 56.75%. For men, follow the same process to find the percentage, which is approximately 4.65%.
Step-by-step explanation:
In order to find the percentage of women meeting the height requirement, we need to calculate the area under the curve of the normal distribution for women with heights between 50 inches and 64 inches. This can be done by finding the z-scores for these values and using the z-score table or a calculator. The z-score for a height of 50 inches is calculated as follows:
z = (50 - 63.6) / 2.3 = -5.91
Using the z-score table, we can find that the area to the left of z = -5.91 is approximately 0.0000000002. Since this probability is too small, we can assume that no women in the population have a height less than 50 inches. Similarly, we can calculate the z-scores and probabilities for a height of 64 inches and find that the area under the curve to the left of z = 0.17 is approximately 0.5675.
Therefore, the percentage of women meeting the height requirement is approximately 56.75%.
To find the percentage of men meeting the height requirement, we follow the same process. The z-score for a height of 50 inches for men is calculated as follows:
z = (50 - 69.8) / 3.4 = -5.82
Using the z-score table or a calculator, we can find that the area to the left of z = -5.82 is approximately 0.0000000002. This probability is too small, so we can assume that no men in the population have a height less than 50 inches. For a height of 64 inches, the z-score is calculated as:
z = (64 - 69.8) / 3.4 = -1.68
The area to the left of z = -1.68 is approximately 0.0465. Therefore, the percentage of men meeting the height requirement is approximately 4.65%.