Suppose that the heights of female adults in the US are normally distributed with a mean (µ) of 65.4 inches and a standard deviation (σ) of 2.8 inches. Let X denote the height of a randomly chosen adult female. Find the probability that X is between 66 and 70 inches:
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.365
To find the probability that X is between 66 and 70 inches, we can use the given mean and standard deviation to standardize the values and then look up the corresponding probabilities in the standard normal distribution table.
Step 1: Standardize the values
To standardize a value, we use the formula:
z = (X - µ) / σ
where z is the standardized value, X is the original value, µ is the mean, and σ is the standard deviation.
For X = 66 inches:
z1 = (66 - 65.4) / 2.8 = 0.214
For X = 70 inches:
z2 = (70 - 65.4) / 2.8 = 1.643
Step 2: Look up the probabilities
Now, we look up the probabilities corresponding to the standardized values z1 and z2 in the standard normal distribution table.
From the table, we find:
P(Z < 0.214) = 0.5864 (approximately)
P(Z < 1.643) = 0.9500 (approximately)
Step 3: Calculate the desired probability
To find the probability that X is between 66 and 70 inches, we subtract the probability corresponding to z1 from the probability corresponding to z2:
P(66 ≤ X ≤ 70) = P(Z < 1.643) - P(Z < 0.214)
= 0.9500 - 0.5864
= 0.3636 (approximately)
Therefore, the probability that X is between 66 and 70 inches is approximately 0.3636.