Jim Tree wants to analyze his shipment of trees based on height. He knows the height of the trees is normally distributed so he
can use the standard normal distribution. He measures the height of 100 randomly selected trees in his shipment. He finds the mean is 65 inches and the standard deviation is 10 inches. What percentage of the trees will be between 55 inches and 75 inches?
Answer: If the mean is 65 inches, and the standard deviation is 10 inches, then when the problem ask for the percentage of the trees between 55 inches and 75 inches, is actually asking: which percentage of the trees is between the mean ± standard variation, or 65 inches ± 10 inches.
We know that if a distribution is normal, then the standard deviation encloses 34.1% of the set. But in this case, we are enclosing the area for ± standard variation (this is two times the standard deviation, one for the left and one for the right) then you are enclosing 34.1% two times, and this is 68.2%.
A 68.2% of the trees is between 55 inches and 75 inches.
