The distribution of the heights of men in the U.S. is normally distributed with a mean of 70 inches and a standard deviation of 5 inches.
a) What is the probability of an american male being less than 60 inches tall?
b) What is the probability of an american male being between 68 and 72 inches tall?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
To solve these problems, we will use the concept of the standard normal distribution. The standard normal distribution has a mean of 0 and a standard deviation of 1. We will convert the given measurements into standard units and then use a standard normal distribution table or calculator to find the probabilities.
a) To find the probability of an American male being less than 60 inches tall, we need to find the z-score corresponding to 60 inches and then look up the corresponding probability.
The formula for finding the z-score is:
z = (x - mean) / standard deviation
Substituting the given values:
z = (60 - 70) / 5
z = -2
Using a standard normal distribution table or calculator, we find that the probability corresponding to z = -2 is approximately 0.0228. Therefore, the probability of an American male being less than 60 inches tall is approximately 0.0228 or 2.28%.
b) To find the probability of an American male being between 68 and 72 inches tall, we need to find the z-scores corresponding to 68 and 72 inches and then calculate the area between those two z-scores.
The z-score for 68 inches is:
z1 = (68 - 70) / 5
z1 = -0.4
The z-score for 72 inches is:
z2 = (72 - 70) / 5
z2 = 0.4
Using a standard normal distribution table or calculator, we can find the probabilities corresponding to z1 and z2. The probability corresponding to z1 = -0.4 is approximately 0.3446 and the probability corresponding to z2 = 0.4 is also approximately 0.3446.
To calculate the probability of being between 68 and 72 inches tall, we subtract the probability corresponding to z1 from the probability corresponding to z2:
P(68 ≤ X ≤ 72) = P(z1 ≤ Z ≤ z2) = P(Z ≤ 0.4) - P(Z ≤ -0.4)
= 0.3446 - 0.3446
= 0
Therefore, the probability of an American male being between 68 and 72 inches tall is approximately 0 or 0%.