A survey found that women's heights are normally distributed with mean 62.4 in and
standard deviation 2.8 in. The survey also found that men's heights are normally
distributed with a mean of 68.9 in. and standard deviation 2.8. Complete parts a
through c below.
a. Most of the live characters at an amusement park have height requirements with a minimum of 4 ft 9 in. and a maximum of 6 ft 4 in. Find the percentage of women meeting height requirement. The percentage of women who meet the height
requirement is___%. b. Find the percentage of men meeting the height requirement.
The percentage of men who meet the height requirement is____%.
c. If the height requirements are changed to exclude only the tallest 5% of men and
the shortest 5% of women, what are the new height requirements? The new height
requirements are a least___in. and at most___in.
To solve this problem, we need to use the properties of the normal distribution and the given mean and standard deviation for women's and men's heights.
a. To find the percentage of women meeting the height requirement (minimum of 4 ft 9 in. and maximum of 6 ft 4 in.), we can convert these heights into inches.
1 foot = 12 inches
4 ft 9 in = (4 * 12) + 9 = 57 inches
6 ft 4 in = (6 * 12) + 4 = 76 inches
Now, we need to determine the percentage of women whose heights fall within this range. We can use the standard normal distribution table or a statistical calculator to find this percentage.
Using the standard normal distribution table, we need to calculate the z-scores for the minimum and maximum heights for women:
For the minimum height (57 inches):
Z-score = (57 - 62.4) / 2.8 ≈ -1.90
From the standard normal distribution table, the area to the left of -1.90 is approximately 0.0287.
For the maximum height (76 inches):
Z-score = (76 - 62.4) / 2.8 ≈ 4.86
From the standard normal distribution table, the area to the left of 4.86 is approximately 1.0000.
To find the percentage of women meeting the height requirement, we subtract the area to the left of the minimum height from the area to the left of the maximum height:
Percentage of women meeting the height requirement = (1.0000 - 0.0287) * 100 ≈ 97.13%
Therefore, approximately 97.13% of women meet the height requirement.
b. To find the percentage of men meeting the height requirement (minimum of 4 ft 9 in. and maximum of 6 ft 4 in.), we follow the same steps as in part a, but now we use the mean and standard deviation for men's heights.
Using the standard normal distribution table:
For the minimum height (57 inches):
Z-score = (57 - 68.9) / 2.8 ≈ -4.21
From the standard normal distribution table, the area to the left of -4.21 is approximately 0.0000.
For the maximum height (76 inches):
Z-score = (76 - 68.9) / 2.8 ≈ 2.50
From the standard normal distribution table, the area to the left of 2.50 is approximately 0.9938.
Percentage of men meeting the height requirement = (0.9938 - 0.0000) * 100 ≈ 99.38%
Therefore, approximately 99.38% of men meet the height requirement.
c. If the height requirements are changed to exclude only the tallest 5% of men and the shortest 5% of women, we need to find the corresponding heights for these percentiles.
For women:
We want to find the height below which 5% of women fall. From the standard normal distribution table, we look for the Z-score that corresponds to the cumulative area of 0.05 from the left side of the distribution. This Z-score is approximately -1.645.
Using the formula for Z-score:
Z-score = (x - mean) / standard deviation
Rearranging the formula to solve for x:
x = mean + (Z-score * standard deviation)
x = 62.4 + (-1.645 * 2.8) ≈ 57.35 inches
Therefore, the new minimum height requirement for women is approximately 57.35 inches.
For men:
We want to find the height above which 5% of men fall. From the standard normal distribution table, we look for the Z-score that corresponds to the cumulative area of 0.95 from the left side of the distribution. This Z-score is approximately 1.645.
Using the formula for Z-score:
Z-score = (x - mean) / standard deviation
Rearranging the formula to solve for x:
x = mean + (Z-score * standard deviation)
x = 68.9 + (1.645 * 2.8) ≈ 73.56 inches
Therefore, the new maximum height requirement for men is approximately 73.56 inches.
The new height requirements are at least 57.35 inches and at most 73.56 inches.