The distribution of heights of American women aged 18 to 24 is approximately normally distributed with mean 65.5 inches and standard deviation 2.5 inches. What is the probability that a randomly selected woman is between 60 and 64 inches tall?
To find the probability that a randomly selected woman is between 60 and 64 inches tall, we need to use the normal distribution.
Step 1: Standardize the values
First, we need to standardize the values of 60 and 64 inches using the formula:
z = (x - μ) / σ
where:
- z is the standard score (z-score)
- x is the observed value
- μ is the mean
- σ is the standard deviation
For 60 inches:
z_60 = (60 - 65.5) / 2.5 = -2.2
For 64 inches:
z_64 = (64 - 65.5) / 2.5 = -0.6
Step 2: Calculate the area under the normal curve
Next, we need to find the area under the normal curve between these two standard scores. Since the normal distribution is symmetric, we can use the z-table or a statistical software to find these probabilities.
Using the z-table, we can find the area to the left of z_64 and z_60 and then subtract them to find the probability between the two values.
The z-table gives us:
- z_60 area = 0.0139
- z_64 area = 0.2743
Step 3: Calculate the probability
To find the probability between 60 and 64 inches, we subtract the area to the left of 60 inches from the area to the left of 64 inches:
P(60 ≤ X ≤ 64) = P(X ≤ 64) - P(X ≤ 60)
= 0.2743 - 0.0139
= 0.2604
Therefore, the probability that a randomly selected woman is between 60 and 64 inches tall is approximately 0.2604 or 26.04%.