The average height for Chinese women is 62.5 inches with a standard deviation of 4.6 inches. What is the probability that a randomly selected woman in the United States would be ≥69 inches tall?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
To calculate the probability that a randomly selected woman in the United States would be at least 69 inches tall, we need to standardize the height using z-score and then find the corresponding probability.
Step 1: Calculate the z-score.
The z-score formula is given by:
z = (x - μ) / σ
where:
x = the value (height) we want to standardize
μ = the mean (average height for Chinese women) = 62.5 inches
σ = the standard deviation = 4.6 inches
Plugging in the values:
z = (69 - 62.5) / 4.6
z = 1.41
Step 2: Find the probability.
We need to find the probability of a z-score greater than 1.41. Since we want to know the probability that a woman is at least 69 inches tall, we need to find the probability of the z-score being greater than 1.41 (since it corresponds to a height greater than or equal to 69 inches).
Using a standard normal distribution table, we can find the probability associated with a z-score of 1.41.
The probability corresponding to a z-score of 1.41 is approximately 0.9207.
Therefore, the probability that a randomly selected woman in the United States would be at least 69 inches tall is approximately 0.9207 or 92.07%.
To find the probability that a randomly selected woman in the United States would be 69 inches tall or taller, we can use the standard normal distribution.
First, we need to calculate the z-score, which measures how many standard deviations an individual value is from the average. We can calculate the z-score using the formula:
z = (x - μ) / σ
Where:
x is the value we want to find the probability for (69 inches),
μ is the mean (average) height for Chinese women (62.5 inches), and
σ is the standard deviation of the height for Chinese women (4.6 inches).
Plugging in the values, we get:
z = (69 - 62.5) / 4.6 ≈ 1.413
Next, we need to find the corresponding probability using a standard normal distribution table or a calculator. The table or calculator will give us the probability to the left of the z-score. Since we want the probability for values greater than or equal to 69 inches, we need to find the probability to the right of the z-score. We can obtain this by subtracting the probability to the left of the z-score from 1.
Using a standard normal distribution table, we find that the probability to the left of a z-score of 1.413 is approximately 0.9211. Subtracting this value from 1, we get:
P(z ≥ 1.413) ≈ 1 - 0.9211 ≈ 0.0789
Therefore, the probability that a randomly selected woman in the United States is 69 inches tall or taller is approximately 0.0789, or 7.89%.
Remember that this probability assumes that the height of women in the United States follows a normal distribution with the same mean and standard deviation as the Chinese women's height. It's worth noting that this assumption may or may not be accurate in reality.