Final answer:
a. Sketch the Normal curve with mean and standard deviation labeled. b. About 97.72% of young women have heights less than 69.5 inches. c. A young woman with a height of 62 inches is not unusually short.
Step-by-step explanation:
a. To sketch the Normal curve that approximates the distribution of young women's height, we need to draw a bell-shaped curve. The mean is 64.5 inches and the standard deviation is 2.5 inches. We label the mean and the points that are 1, 2, and 3 standard deviations from the mean. The points are: mean (64.5), 1 standard deviation below the mean (64.5 - 2.5 = 62), 1 standard deviation above the mean (64.5 + 2.5 = 67), 2 standard deviations below the mean (64.5 - 2 * 2.5 = 59.5), 2 standard deviations above the mean (64.5 + 2 * 2.5 = 69.5), 3 standard deviations below the mean (64.5 - 3 * 2.5 = 57), and 3 standard deviations above the mean (64.5 + 3 * 2.5 = 72).
b. To find the percent of young women with heights less than 69.5 inches, we need to calculate the z-score first. The z-score formula is: z = (x - µ) / σ. Plugging in the values, we get z = (69.5 - 64.5) / 2.5 = 2. The area to the left of a z-score of 2 is approximately 0.9772 or 97.72%. So, about 97.72% of young women have heights less than 69.5 inches.
c. To determine if a young woman with a height of 62 inches is unusually short, we need to check the z-score. The z-score formula is: z = (x - µ) / σ. Plugging in the values, we get z = (62 - 64.5) / 2.5 = -1. The area to the left of a z-score of -1 is approximately 0.1587 or 15.87%. Since 62 inches falls within 1 standard deviation below the mean, which is a common range, it can be considered within the normal range and not unusually short.