Question 1171440: The study about the height of Southeast Asian women follows a normal distribution with a mean of 153 cm and standard deviation 12 cm. What is the probability that a randomly selected woman’s height is less than 150 cm?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem using the properties of the normal distribution.
**1. Define the Variables**
* Mean (μ) = 153 cm
* Standard deviation (σ) = 12 cm
* We want to find the probability that a woman's height (X) is less than 150 cm, i.e., P(X < 150).
**2. Calculate the Z-score**
The Z-score represents how many standard deviations a value is from the mean. It's calculated using the formula:
* Z = (X - μ) / σ
In our case:
* Z = (150 - 153) / 12
* Z = -3 / 12
* Z = -0.25
**3. Find the Probability Using the Z-table or Calculator**
We need to find the probability P(Z < -0.25). This is the area under the standard normal distribution curve to the left of Z = -0.25.
You can find this probability using:
* A standard normal distribution table (Z-table)
* A calculator with statistical functions
* An online normal distribution calculator
Using a Z-table or calculator, you will find:
* P(Z < -0.25) ≈ 0.4013
**4. Interpret the Result**
The probability that a randomly selected Southeast Asian woman's height is less than 150 cm is approximately 0.4013, or 40.13%.
**Therefore, the probability is approximately 0.4013.**
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