A survey found that women's heights are normally distributed with mean 62.5
in. and standard deviation 2.7
in. The survey also found that men's heights are normally distributed with a mean 67.9
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 2
in. Find the percentage of women meeting the height requirement.
The percentage of women who meet the height requirement is nothing
%.
(Round to two decimal places as needed.)
b. Find the percentage of men meeting the height requirement.
The percentage of men who meet the height requirement is nothing
%.
(Round to two decimal places as needed.)
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 at least nothing
in. and at most nothing
in.
(Round to one dec
a. To find the percentage of women meeting the height requirement, we need to find the area under the normal distribution curve between 4 ft 9 in. and 6 ft 2 in.
First, we need to convert the heights to inches. 4 ft 9 in. is equal to (4 x 12) + 9 = 57 inches, and 6 ft 2 in. is equal to (6 x 12) + 2 = 74 inches.
Next, we can standardize these values using the z-score formula:
z1 = (57 - 62.5) / 2.7
z2 = (74 - 62.5) / 2.7
Using a z-table or a calculator, we can find the area under the normal distribution curve between z1 and z2. This represents the percentage of women meeting the height requirement.
b. Similarly, to find the percentage of men meeting the height requirement, we need to find the area under the normal distribution curve between 4 ft 9 in. and 6 ft 2 in., but using the mean and standard deviation for men's heights.
c. To find the new height requirements, we need to find the heights that correspond to the top 5% and bottom 5% of the normal distribution curve for men and women respectively. This can be done by finding the z-scores that correspond to the cumulative probabilities of 0.05 and 0.95, and then using the z-score formula to find the corresponding heights.
Note: Without specific values for z-scores corresponding to the cumulative probabilities of 0.05 and 0.95, we cannot provide the exact new height requirements.
To find the percentage of women meeting the height requirement, we need to calculate the area under the normal distribution curve between 4 ft 9 in (which is equivalent to 57 inches) and 6 ft 2 in (which is equivalent to 74 inches) in terms of standard deviations.
1. Convert the height requirements to z-scores:
For the lower height requirement: (57 - 62.5) / 2.7 = -2.04
For the upper height requirement: (74 - 62.5) / 2.7 = 4.26
2. Use a standard normal distribution table or a calculator to find the area between -2.04 and 4.26. This represents the percentage of women meeting the height requirement.
The percentage of women who meet the height requirement is given by the area between -2.04 and 4.26 under the standard normal distribution curve. The z-score table indicates that the area to the left of -2.04 is approximately 0.0207 and the area to the left of 4.26 is approximately 0.9997. Thus, the area between -2.04 and 4.26 is approximately 0.9997 - 0.0207 = 0.979.
Therefore, the percentage of women meeting the height requirement is 0.979 * 100 = 97.9%. Rounded to two decimal places, it is 97.90%.
To find the percentage of men meeting the height requirement, we follow the same steps:
1. Convert the height requirements to z-scores:
For the lower height requirement: (57 - 67.9) / 2.8 = -3.89
For the upper height requirement: (74 - 67.9) / 2.8 = 2.18
2. Use a standard normal distribution table or a calculator to find the area between -3.89 and 2.18. This represents the percentage of men meeting the height requirement.
The z-score table indicates that the area to the left of -3.89 is approximately 0.0000459 and the area to the left of 2.18 is approximately 0.9857. Thus, the area between -3.89 and 2.18 is approximately 0.9857 - 0.0000459 = 0.9856.
Therefore, the percentage of men meeting the height requirement is 0.9856 * 100 = 98.56%. Rounded to two decimal places, it is 98.56%.
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 height values.
To find the cutoff height for the tallest 5% of men, we need to find the z-score that corresponds to an area of 0.95 (1 - 0.05) under the standard normal distribution curve. Using a z-score table or calculator, we find that the z-score corresponding to an area of 0.95 is approximately 1.645.
The cutoff height for men is therefore (1.645 * 2.8) + 67.9 = 72.768 inches. Rounded to one decimal place, it is 72.8 inches.
To find the cutoff height for the shortest 5% of women, we need to find the z-score that corresponds to an area of 0.05 under the standard normal distribution curve.
Using a z-score table or calculator, we find that the z-score corresponding to an area of 0.05 is approximately -1.645.
The cutoff height for women is therefore (-1.645 * 2.7) + 62.5 = 58.758 inches. Rounded to one decimal place, it is 58.8 inches.
Therefore, the new height requirements are at least 58.8 inches and at most 72.8 inches.
recipe is here:
http://davidmlane.com/hyperstat/z_table.html