It is estimated that heights of adult men are normally distributed with a mean of 70 inches and a standard deviation of 3.5. In one state, the law requires a person to be 68 inches or taller to become a fire fighter.
(a) What proportion of adult men will meet this height requirement for becoming a fire fighter in this state?
(b) A person who was denied to become a fire fighter learned that his height was at the 20th percentile. What was his actual height?
a. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
b. Use same table to find Z and insert into equation above to solve.
To solve this problem, we will use the concept of standard normal distribution.
(a) To find the proportion of adult men who meet the height requirement of 68 inches or taller, we need to calculate the area under the curve of the standard normal distribution to the right of 68 inches.
Step 1: Standardize the height requirement using the formula z = (x - μ) / σ
where x is the height requirement (68 inches), μ is the mean (70 inches), and σ is the standard deviation (3.5 inches).
z = (68 - 70) / 3.5
z = -2 / 3.5
z ≈ -0.5714
Step 2: Find the area to the right of z = -0.5714 using a standard normal distribution table or calculator. This area represents the proportion of adult men who meet the height requirement.
Consulting a standard normal distribution table or using a calculator, we find that the area to the right of z = -0.5714 is approximately 0.7157.
Therefore, approximately 71.57% of adult men will meet the height requirement for becoming a fire fighter in this state.
(b) To find the height of a person who was denied to become a fire fighter and whose height corresponds to the 20th percentile, we need to find the z-score corresponding to the 20th percentile and then convert it back to the actual height.
Step 1: Determine the z-score corresponding to the 20th percentile using a standard normal distribution table or calculator. The 20th percentile corresponds to an area to the left of z.
Consulting a standard normal distribution table or using a calculator, we find that the z-score corresponding to the 20th percentile is approximately -0.8416.
Step 2: Convert the z-score back to the actual height using the formula:
x = z * σ + μ
where x is the actual height, z is the z-score, σ is the standard deviation (3.5 inches), and μ is the mean (70 inches).
x = -0.8416 * 3.5 + 70
x ≈ 66.03
Therefore, the actual height of the person who was denied to become a fire fighter is approximately 66.03 inches.