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Answer :
Answer:
a) 68% of the men fall between 169 cm and 183 cm of height.
b) 95% of the men will fall between 162 cm and 190 cm.
c) It is unusual for a man to be more than 197 cm tall.
Step-by-step explanation:
The 68-95-99.5 empirical rule can be used to solve this problem.
This values correspond to the percentage of data that falls within in a band around the mean with two, four and six standard deviations of width.
a) What is the approximate percentage of men between 169 and 183 cm?
To calculate this in an empirical way, we compare the values of this interval with the mean and the standard deviation and can be seen that this interval is one-standard deviation around the mean:
[tex]\mu-\sigma=176-7=169\\\mu+\sigma=176+7=183[/tex]
Empirically, for bell-shaped distributions and approximately normal, it can be said that 68% of the men fall between 169 cm and 183 cm of height.
b) Between which 2 heights would 95% of men fall?
This corresponds to ±2 standard deviations off the mean.
[tex]\mu-2\sigma=176-2*7=162\\\\\mu+2\sigma=176+2*7=190[/tex]
95% of the men will fall between 162 cm and 190 cm.
c) Is it unusual for a man to be more than 197 cm tall?
The number of standard deviations of distance from the mean is
[tex]n=(197-176)/7=3[/tex]
The percentage that lies outside 3 sigmas is 0.5%, so only 0.25% is expected to be 197 cm.
It can be said that is unusual for a man to be more than 197 cm tall.
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Final answer:
Using the Empirical Rule, 68% of men will have a height between 169 and 183 cm, 95% of men should fall between the heights of 162 cm and 190 cm, and a man taller than 197 cm would be considered unusual.
Explanation:
The given data mentions that the mean height of men is 176 cm with a standard deviation of 7 cm. The Empirical Rule, also known as the 68-95-99.7 rule, comes into play here.
a) Percentage of men between 169 and 183 cm.
When we add or subtract one standard deviation from the mean (176 + 7 = 183, 176 - 7 = 169), it accounts for approximately 68% of the population. So, about 68% of men will have a height between 169 and 183 cm.
b) Heights representing 95% of men.
Two standard deviations from the mean (176 + 2*7 = 190, 176 - 2*7 = 162) accounts for about 95% of the population. So, 95% of men should fall between the heights of 162 cm and 190 cm.
c) Unusual height greater than 197 cm.
Three standard deviations from the mean (176 + 3*7 = 197) capture about 99.7% of the population. Hence, a man taller than 197 cm would be considered unusual as it exceeds the range considered for 99.7% of the population.
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