Final answer:
The empirical rule is used to determine the percentage of women's heights falling within certain ranges around the mean of 63.5 inches and standard deviation of 2.5 inches. For one standard deviation (61-66 inches), the percentage is 68%; for two standard deviations (56-71 inches), the percentage is 95%; for half above the mean to one standard deviation (63.5-66 inches), the percentage is 34%; less than the mean is 50%; more than 58.5 inches is 66%.
Step-by-step explanation:
Using the empirical rule (68-95-99.7), which applies to normally distributed data, we can answer the given questions about women's heights in the U.S. The heights are normally distributed with a mean (μ) of 63.5 inches and a standard deviation (σ) of 2.5 inches. Moreover, the empirical rule states that approximately 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
- a. The percentage of women between 61 and 66 inches tall is the percentage within one standard deviation from the mean, which is 68%.
- b. The percentage of women between 56 and 71 inches tall represents the percentage within two standard deviations from the mean, amounting to 95%.
- c. The percentage of women between 63.5 (the mean) and 66 inches tall is half of 68%, as this range captures the middle to one standard deviation above the mean, which is 34%.
- d. The percentage of women less than 63.5 inches tall is 50%, which is the lower half of the distribution.
- e. The percentage of women more than 58.5 inches tall includes all women within three standard deviations above the lower extreme, encompassing virtually 99.7% of the distribution. Since 58.5 is one standard deviation below the mean, we subtract the percentage within one standard deviation below the mean (34%) from 100%, resulting in 100% - 34% = 66% of women being more than 58.5 inches tall.