Tall clubs international is a social organization for tall people. It has a height requirement that men must be at least 74 in. and women must be at least 70 in. tall. If men's heights have a normal mean of 69.0 in and a deviation of 2.8 and women's normal heights have a mean of 63.6 in. and a deviation of 2.5 in.
a) what percentage of men meet that requirement?
b) what percentage of women meet that requirement?
C)Are the requirements for men and women fair? why and why not?
z = (score -mean)/sd
a. z = (74-69)/2.8
z = 5/2.8 = 1.79 is .0367
The percentage of men that are taller than in is 3.67%
b. z = (score -mean)/sd
z = ((70-63.6)/2.5
z = 6.4/2.5 = 2.56 is .0052
The percentage of women that are taller than in is .52%.
To find the percentage of men who meet the height requirement, we need to calculate the proportion of men whose height is at least 74 inches. Since we have the mean and standard deviation of men's heights, we can use the standard normal distribution to make this calculation.
a) To find the percentage of men meeting the requirement:
Step 1: Calculate the z-score for the height requirement using the formula: z = (x - mean) / standard deviation
For men: z = (74 - 69) / 2.8 = 1.7857 (rounded to 4 decimal places)
Step 2: Use a z-table or calculator to find the percentage of values below the z-score of 1.7857. According to the standard normal distribution table, this z-score corresponds to approximately 0.9633 (or 96.33%) below it.
Therefore, approximately 96.33% of men meet the height requirement.
b) Following a similar process as for men:
Step 1: Calculate the z-score for the height requirement for women:
z = (70 - 63.6) / 2.5 = 2.56 (rounded to 2 decimal places)
Step 2: Use the z-table or calculator to find the percentage of values below the z-score of 2.56. According to the standard normal distribution table, this z-score corresponds to approximately 0.9943 (or 99.43%) below it.
Therefore, approximately 99.43% of women meet the height requirement.
c) In terms of fairness, we can assess the requirements for men and women by comparing the proportions who meet the respective height requirements. Based on the calculations, a higher percentage of women (99.43%) meet the height requirement compared to men (96.33%). This suggests that the requirements may not be fair because it sets a higher standard for men relative to women. The requirements overemphasize tallness for men compared to women.
To determine the percentage of men and women that meet the height requirement for Tall Clubs International, we need to calculate the z-scores for both men and women and then use the standard normal distribution table.
a) Percentage of men who meet the height requirement:
First, let's calculate the z-score for the minimum height requirement for men, which is 74 inches:
z = (X - μ) / σ
z = (74 - 69.0) / 2.8
z = 1.7857
Now, let's refer to the standard normal distribution table to find the percentage of men that fall above this z-score. Looking up 1.7857 in the table, we find that the corresponding area is approximately 0.9633.
Since we want the percentage of men who meet the height requirement (which is those falling above the z-score), we subtract the area from 1 (100%):
Percentage of men who meet the requirement = 1 - 0.9633 ≈ 0.0367 or 3.67%
Therefore, approximately 3.67% of men meet the height requirement for Tall Clubs International.
b) Percentage of women who meet the height requirement:
Similarly, we need to calculate the z-score for the minimum height requirement for women, which is 70 inches:
z = (X - μ) / σ
z = (70 - 63.6) / 2.5
z = 2.56
Using the standard normal distribution table, we find that the area to the left of 2.56 is approximately 0.9948.
To find the percentage of women that meet the height requirement (those falling below the z-score), we multiply the area by 100%:
Percentage of women who meet the requirement = 0.9948 × 100 ≈ 99.48%
Therefore, approximately 99.48% of women meet the height requirement for Tall Clubs International.
c) Fairness of the height requirements:
Based on the calculated percentages, it seems that the height requirement is more challenging for men compared to women. Only around 3.67% of men meet the requirement, while almost 99.48% of women meet the requirement.
This raises questions about the fairness of the requirements. Since the required height for men follows a normal distribution with a mean of 69.0 inches and a deviation of 2.8, it seems much more difficult for men to meet the height criteria. On the other hand, the required height for women follows a normal distribution with a mean of 63.6 inches and a deviation of 2.5, making it easier for women to meet the criteria.
Whether these requirements are fair or not is subjective and depends on the purpose and goals of the organization. If the aim is simply to create a space for taller individuals to connect, then the height requirements may not be fair. However, if the goal is to provide a social organization specifically for exceptionally tall individuals, then the requirements may be considered fair in that context.