Question

A survey found that women's heights are normally distributed with mean 63.4 in and standard deviation 3.7 in. The survey also found that men's heights are normally distributed with mean 68.4 in and standard deviation 3.2 in. Most of the live characters employed an amusement park have height requirements of a minimum 55 in and a maximum of 64 in.

Complete parts (a) and (b) below

A) find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement park? The percentage of men who meet the height requirement is ____ %

B. If the height requirements are changed to exclude only the tallest 50% of men and the shortest 5% of men, what are the new height requirements? The new height requirements are a minimum of ___ in and a maximum of ___ in

Answer 1

The percentage of men meeting the original height requirement is low due to their generally taller stature in comparison to the requirement range. When adjusting requirements to exclude the tallest 50% and shortest 5% of men, only the new minimum height is calculated, as the maximum height remains unchanged.

Step-by-step explanation:

Part A: Percentage of Men Meeting Height Requirement

To find the percentage of men who meet the height requirement for the live characters employed at an amusement park, we must calculate the area under the normal distribution curve for men's heights between the minimum of 55 inches and the maximum of 64 inches. Considering the mean height for men is 68.4 inches, with a standard deviation of 3.2 inches, we calculate the z-scores for both height limits.

Z for 55 inches = (55 - 68.4) / 3.2 = -4.19

Z for 64 inches = (64 - 68.4) / 3.2 = -1.38

Using a z-table or normal distribution calculator, we find the probabilities corresponding to these z-scores and subtract the smaller from the larger to find the percentage of men within the range.

This result suggests that a smaller percentage of men are likely to be employed as characters at the amusement park due to the height requirement when compared to women.

Part B: New Height Requirements for Men

To exclude only the tallest 50% and the shortest 5% of men, we look up the z-scores corresponding to the 50th percentile (which is the median, so a z-score of 0) and the 5th percentile on the normal distribution curve. The z-score for the 5th percentile is approximately -1.645.

So, the new height minimum is the mean plus the z-score for the 5th percentile times the standard deviation:

New minimum height = 68.4 + (-1.645 * 3.2) inches.

The maximum height is not changed as it will be higher than the mean, and the current requirement only excludes individuals higher than the mean (top 50%).