Step 1 :The problem is asking for the probability of selecting a woman with a height less than 66.4 inches. This is a problem of normal distribution.
Step 2 :We can calculate the z-score for the height 66.4 inches and then find the corresponding probability from the standard normal distribution table.
Step 3 :The z-score is calculated as \( (X - \mu) / \sigma \), where X is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 4 :In this case, \(X = 66.4\), \(\mu = 64.2\), and \(\sigma = 2.7\).
Step 5 :Calculate the z-score: \( z = (X - \mu) / \sigma = (66.4 - 64.2) / 2.7 = 0.8148 \).
Step 6 :Use the cumulative distribution function to find the probability corresponding to the z-score.
Step 7 :The probability of randomly selecting 1 woman with a height less than 66.4 inches is approximately 0.7924.
Step 8 :Final Answer: The probability of randomly selecting 1 woman with a height less than 66.4 inches is \(\boxed{0.7924}\).