Final answer:
To calculate the probability of a woman's height being between 62 and 70 inches, we use the normal distribution and z-scores to find the corresponding probabilities from a z-table or software, and then subtract to get the desired probability.
Step-by-step explanation:
The question seeks to find the probability that a randomly selected woman is between 62 inches and 70 inches in height, given that the mean height is 65 inches and the standard deviation is 2.7 inches. This question can be solved using the concept of the normal distribution and z-scores.
First, calculate the z-score for both 62 inches and 70 inches using the formula z = (X - µ) / σ, where 'X' is the height, 'µ' is the mean and 'σ' is the standard deviation. After finding the z-scores for both heights, the next step is to refer to the standard normal distribution table (or use a statistical software) to find the probabilities corresponding to these z-scores. Finally, the probability that a woman's height falls between the two heights is found by subtracting the probability of the lower height from that of the higher height.
It should be noted that the exact solution requires calculations and the use of a z-table or software to find the probabilities, but the question itself aligns with understanding the application of normal distribution in probability and statistics.