The heights of American women ages 18 to 29 are normally distributed with a mean of 64.3 inches and a standard deviation of 3.8 inches. An American woman in this age bracket is chosen at random. What is the probability that she is less than 70 inches tall?
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The heights of American women ages 18 to 29 are normally distributed with a mean of 64.3 inches and a standard deviation of 3.8 inches. An American woman in this age bracket is chosen at random. What is the probability that she is less than 70 inches tall?
To find the probability that an American woman in this age bracket is less than 70 inches tall, we need to calculate the cumulative probability using the given mean and standard deviation.
Step 1: Calculate the z-score:
The z-score formula is given by:
z = (x - μ) / σ
where:
x = 70 inches (the value we are interested in)
μ = 64.3 inches (mean)
σ = 3.8 inches (standard deviation)
Plugging in the values, we get:
z = (70 - 64.3) / 3.8
Step 2: Find the cumulative probability:
Using a standard normal distribution table or a calculator, find the cumulative probability associated with the z-score. This will give us the probability of being less than the given value.
Let's assume the cumulative probability associated with the z-score is P(z < z-value).
Therefore, the probability of being less than 70 inches is P(x < 70) = P(z < z-value).
Step 3: Calculate the probability:
Look up the z-value in the standard normal distribution table or use a calculator to find the cumulative probability associated with the z-value.
Let's assume the cumulative probability associated with the z-value is 0.9072.
Therefore, the probability that an American woman in this age bracket is less than 70 inches tall is 0.9072 or 90.72%.
Note: The actual z-value and cumulative probability may differ depending on the accuracy of the calculation method used.
To find the probability that an American woman in the given age bracket is less than 70 inches tall, you can use the information provided about the height distribution, mean, and standard deviation.
Step 1: Determine the Z-score
The first step is to calculate the Z-score, which measures how many standard deviations the value is away from the mean.
Z = (x - μ) / σ
Where:
Z is the Z-score
x is the given value (70 inches)
μ is the mean (64.3 inches)
σ is the standard deviation (3.8 inches)
Z = (70 - 64.3) / 3.8
Z = 1.5
Step 2: Find the probability using the Z-score
Next, you can use a standard normal distribution (also known as a Z-table or a Z-score table) to find the probability corresponding to the Z-score of 1.5.
Consulting the Z-table, you can find that the cumulative probability (area under the curve) to the left of 1.5 is approximately 0.9332.
Step 3: Calculate the final probability
The final step is to subtract the cumulative probability from 1, since you are looking for the probability that the height is less than 70 inches.
P(X < 70) ≈ 1 - 0.9332
P(X < 70) ≈ 0.0668
Therefore, the probability that an American woman in the given age bracket is less than 70 inches tall is approximately 0.0668, or 6.68%.