Answer :
Final answer:
According to the empirical rule:
(a) Approximately 68% of English men are between 62.7 and 75.3 inches tall.
(b) Approximately 68% of English men are under 66.9 inches tall.
(c) Approximately 95% of English men are over 64.8 inches tall.
Explanation:
To calculate the percentage of English men falling within specific height ranges, we can use the empirical rule. Let's calculate each part:
(a) Between 62.7 and 75.3 inches tall:
First, we need to find the number of standard deviations away from the mean for each boundary:
Lower boundary: (62.7 - 69) / 2.1 = -3.1 / 2.1 = -1.476
Upper boundary: (75.3 - 69) / 2.1 = 6.3 / 2.1 = 3
According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean. Since the range from -1.476 to 3 covers more than one standard deviation, we can conclude that the percentage of English men between 62.7 and 75.3 inches tall is approximately 68%.
(b) Under 66.9 inches tall:
To find the number of standard deviations away from the mean for 66.9 inches:
(66.9 - 69) / 2.1 = -2.1 / 2.1 = -1
According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean. Since -1 is within one standard deviation, we can conclude that the percentage of English men under 66.9 inches tall is approximately 68%.
(c) Over 64.8 inches tall:
To find the number of standard deviations away from the mean for 64.8 inches:
(64.8 - 69) / 2.1 = -4.2 / 2.1 = -2
According to the empirical rule, approximately 95% of the data falls within two standard deviations of the mean. Since -2 is within two standard deviations, we can conclude that the percentage of English men over 64.8 inches tall is approximately 95%.
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