Final answer:
To accommodate 99.5% of men's heights with a normal distribution having a mean of 70 inches and a standard deviation of 3 inches, door frames should be at least 77.74 inches tall, making Option 4: have a height of 79 inches or more, the closest and correct choice.
Step-by-step explanation:
The question revolves around the concept of a normal distribution and determining what height door frames should be to accommodate 99.5% of men given that men's heights are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. To answer this question, we use the properties of the normal distribution curve where about 99.5% of the data falls within a certain range below the mean. This range is determined by the z-score that corresponds to 99.5% in the standard normal distribution. Using the z-score table or a calculator, we find that a z-score of about 2.58 corresponds to the upper 99.5% of a normal distribution. Therefore, to find the cutoff height, we use the formula for a z-score:
Z = (X - mean) / standard deviation
Here, we solve for X, which represents the height:
2.58 = (X - 70) / 3
Now, we multiply both sides by the standard deviation and add the mean:
X = 2.58 * 3 + 70
X ≈ 77.74 inches
Door frames then should have a height of 77.74 inches or more to accommodate 99.5% of men's heights. Of the given options, Option 4: have a height of 79 inches or more is the closest to this number and therefore is the correct choice to ensure 99.5% of men will fit through the door frames without issue.