Final answer:
To find the probability of a woman's height is between 61.8 inches and 62.8 inches, calculate the z-scores for both heights, look up the corresponding probabilities, and subtract the smaller probability from the larger one. You can also use the standard normal distribution and the z-score formula.
Step-by-step explanation:
The student's question involves finding the probability that a randomly selected woman's height falls within a certain interval when women's heights are normally distributed with a mean of 62.2 inches and a standard deviation of 2.8 inches.
Once you have the z-scores, you can find the probability using the cumulative distribution function (CDF) for the standard normal distribution.
Let's denote the random variable representing women's height by X. To find the probability that X is between 61.8 inches and 62.8 inches, we first need to calculate the z-scores corresponding to these heights.
The z-score is given by the formula: z = (X - mean) / standard deviation.
For 61.8 inches, the z-score is: z1 = (61.8 - 62.2) / 2.8 = -0.14
For 62.8 inches, the z-score is: z2 = (62.8 - 62.2) / 2.8 = 0.21
Now, we check a z-table or use statistical software to get the probabilities corresponding to these z-scores. Then we subtract the smaller probability from the larger one to find the probability that a randomly selected woman's height falls between 61.8 inches and 62.8 inches.