According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches.
A. What is the probability that an adult male chosen at random is between 61 inches and 71 inches tall?
B. What is the probability that an adult male chosen at random is more than 6 feet tall?
a good place to play around with Z table stuff is
http://davidmlane.com/hyperstat/z_table.html
Awesome! I got .7603 and .142 as my answer. Thank you!
To find the probability, we will need to use the standard normal distribution.
A. To calculate the probability that an adult male chosen at random is between 61 inches and 71 inches tall, we need to calculate the z-scores for both values and then find the area under the standard normal curve between those z-scores.
The z-score formula is:
z = (x - μ) / σ
Where:
x = the value we are interested in (in this case, the heights of 61 inches and 71 inches)
μ = the mean of the distribution (69.0 inches)
σ = the standard deviation of the distribution (2.8 inches)
For the lower bound:
z1 = (61 - 69) / 2.8 = -2.857
For the upper bound:
z2 = (71 - 69) / 2.8 = 0.714
Using a standard normal distribution table or a calculator, we can find the probabilities corresponding to these z-scores. The probability between these two values can be found by subtracting the probability for the lower bound from the probability for the upper bound:
P(61 ≤ x ≤ 71) = P(z1 ≤ z ≤ z2)
Calculating this using a standard normal distribution table or a calculator will give you the desired probability.
B. To calculate the probability that an adult male chosen at random is more than 6 feet (72 inches) tall, we can calculate the z-score for 72 inches and find the area under the standard normal curve to the right of that z-score:
z = (x - μ) / σ
z = (72 - 69) / 2.8 = 1.071
P(x > 72) = P(z > 1.071)
Again, using a standard normal distribution table or a calculator, you can find the probability corresponding to this z-score.
To find the probabilities in question, we can use the concept of Z-scores and the standard normal distribution table. A Z-score represents how many standard deviations a given value is from the mean.
A. To find the probability that an adult male is between 61 inches and 71 inches tall, we need to find the Z-scores corresponding to these values and then find the area under the normal curve between these Z-scores.
First, we calculate the Z-score for 61 inches:
Z1 = (61 - 69) / 2.8 = -2.857
Next, we calculate the Z-score for 71 inches:
Z2 = (71 - 69) / 2.8 = 0.714
Using the Z-score table or a statistical calculator, we can find the cumulative probability for each Z-score.
For Z1 = -2.857, the cumulative probability is 0.0021.
For Z2 = 0.714, the cumulative probability is 0.7611.
To find the probability between these two Z-scores, we subtract the cumulative probability of Z1 from the cumulative probability of Z2:
P(61 < X < 71) = P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1) = 0.7611 - 0.0021 = 0.759.
Therefore, the probability that an adult male chosen at random is between 61 inches and 71 inches tall is approximately 0.759 or 75.9%.
B. To find the probability that an adult male is more than 6 feet tall, we first need to convert 6 feet (72 inches) into a Z-score.
Z = (X - mean) / standard deviation
Z = (72 - 69) / 2.8 = 1.071
Using the Z-score table or a statistical calculator, we can find the cumulative probability for this Z-score.
For Z = 1.071, the cumulative probability is 0.8577.
However, since we want the probability that an adult male is MORE than 6 feet tall, we subtract this cumulative probability from 1:
P(X > 72) = 1 - P(Z < Z) = 1 - 0.8577 = 0.1423.
Therefore, the probability that an adult male chosen at random is more than 6 feet tall is approximately 0.1423 or 14.23%.