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Answer :
Final answer:
Using the 68-95-99.7 rule, approximately 68% of women's heights are within one standard deviation (65.6 inches to 70.4 inches), about 95% are within two standard deviations (63.2 inches to 72.8 inches), and about 99.7% are within three standard deviations (60.8 inches to 75.2 inches) of the mean height of 68 inches.
Explanation:
According to the 68-95-99.7 rule (also known as the empirical rule), for a normally distributed dataset, approximately:
- 68% of the data falls within one standard deviation of the mean.
- 95% of the data falls within two standard deviations of the mean.
- 99.7% of the data falls within three standard deviations of the mean.
With an average height for women of 68 inches and a standard deviation of 2.4 inches, we can apply this rule to find the following ranges:
- One standard deviation from the mean: 68 ± 2.4 inches, which is between 65.6 and 70.4 inches. This range encompasses about 68% of women's heights.
- Two standard deviations from the mean: 68 ± (2 x 2.4) inches, which is between 63.2 and 72.8 inches. This range encompasses about 95% of women's heights. So, approximately the middle 95% of the women's height will lie within this range.
- Three standard deviations from the mean: 68 ± (3 x 2.4) inches, which is between 60.8 and 75.2 inches. This range encompasses about 99.7% of women's heights.
The typical ranges based on the empirical rule provide a simple way to identify where most of the data in a normally distributed set will lie, in this case for the heights of women in a certain study.
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