the distribution of heights of American women aged 18 to 24 is approximately normally distributed with mean 65.5 inche and standard deviation 2.5 inches. what is the probability that a randomly seleted woman is between 60 and 64 inches tall?
Use z-scores and a z-table to determine your probability.
Formula:
z = (x - mean)/sd
Find both z-values:
z = (60 - 65.5)/2.5
z = (64 - 65.5)/2.5
Once you calculate both z-values, check a z-table for the probability between those two values.
I hope this will help.
To find the probability that a randomly selected woman is between 60 and 64 inches tall, we need to calculate the area under the normal distribution curve between these two values.
Step 1: Standardize the values
First, we need to standardize the values of 60 inches and 64 inches using the formula:
z = (x - mean) / standard deviation
For 60 inches:
z1 = (60 - 65.5) / 2.5
For 64 inches:
z2 = (64 - 65.5) / 2.5
Calculating z1 and z2:
z1 = -2.2
z2 = -0.6
Step 2: Find the probability
Next, we need to find the probability corresponding to the z-scores obtained in step 1. We do this by looking up the values in the standard normal distribution table or using a calculator.
From the z-table, the area to the left of z1 (-2.2) is 0.0139.
From the z-table, the area to the left of z2 (-0.6) is 0.2743.
Step 3: Calculate the probability
To find the probability between 60 and 64 inches, we subtract the probability to the left of z1 from the probability to the left of z2:
P(60 < x < 64) = P(z1 < z < z2) = P(z < z2) - P(z < z1)
P(60 < x < 64) = 0.2743 - 0.0139
P(60 < x < 64) = 0.2604
Therefore, the probability that a randomly selected woman is between 60 and 64 inches tall is approximately 0.2604.
To calculate the probability that a randomly selected woman is between 60 and 64 inches tall, you can use the normal distribution.
First, we need to standardize the values of 60 and 64 using the z-score formula:
z = (x - μ) / σ
where:
- z is the z-score,
- x is the random variable (height),
- μ is the mean,
- σ is the standard deviation.
Given:
- μ = 65.5 inches
- σ = 2.5 inches
- x1 = 60 inches
- x2 = 64 inches
Standardizing x1:
z1 = (60 - 65.5) / 2.5
z1 = -2.2
Standardizing x2:
z2 = (64 - 65.5) / 2.5
z2 = -0.6
Now, we need to find the area under the normal curve between the z-scores z1 and z2. This represents the probability that a randomly selected woman's height is between 60 and 64 inches.
Using a standard normal distribution table or a statistical calculator, you can find the corresponding probabilities for z1 and z2.
P(60 ≤ x ≤ 64) = P(-2.2 ≤ z ≤ -0.6)
By looking at the standard normal distribution table or using a calculator, we find:
P(-2.2 ≤ z ≤ -0.6) ≈ 0.179
Therefore, the probability that a randomly selected woman is between 60 and 64 inches tall is approximately 0.179 or 17.9%.