if the distribution of heights for adult men is approximately normal with a mean of 69.5 inches and a standard deviation of 2.7ninches what is the probability that a randomly selected man is shorter than 65 inches?
To find the probability that a randomly selected man is shorter than 65 inches, we can use the information about the normal distribution with a mean of 69.5 inches and a standard deviation of 2.7 inches.
Step 1: Calculate the z-score
The z-score measures how many standard deviations an individual value is from the mean. In this case, we want to find the z-score for 65 inches.
z = (x - μ) / σ
where:
x = 65 inches (value we're interested in)
μ = 69.5 inches (mean)
σ = 2.7 inches (standard deviation)
Substituting the values into the formula:
z = (65 - 69.5) / 2.7
Step 2: Look up the z-score in the standard normal distribution table
The standard normal distribution table gives the area under the curve to the left of a given z-score. Since we're interested in finding the probability that the randomly selected man is shorter than 65 inches, we need to look up the z-score and find the corresponding area.
Using the z-score of -1.67 (rounded to two decimal places), we can look it up in the standard normal distribution table. The corresponding area is 0.0475.
Step 3: Calculate the probability
The area under the curve to the left of a given z-score represents the probability. In this case, the probability that a randomly selected man is shorter than 65 inches is 0.0475 or 4.75%.
Therefore, the probability that a randomly selected man is shorter than 65 inches is approximately 0.0475 or 4.75%.