68-95-99.7 Rule

Question

Consider that the National Health Center for Statistics reports that females (age 21) have heights that are normally distributed, approximately centered at a mean of 65 inches (5 feet 5 inches) with a standard deviation of 2.5 inches. Use the 68-95-99.7 Rule to find the percentage of females (age 21) with heights in each of the following scenarios:

1. Percentage of heights less than 70: ?%

2. Percentage of heights greater than 67.5: ?%

3. Percentage of heights between 60 and 65: ?%

4. Percentage of heights between 60 and 67.5:?

Answer 1

1. Percentage of heights less than 70: ?%

 

Note that 70% is the value of the mean + 2 standard deviations. Also note that 2 standard deviations contains 95% of the total area or 47.5% above the mean and 47.5% below the mean. We want the total area below 70%, so we want the total area left of the mean, which is 50% (50% is below the mean and 50% is above the mean) + 47.5% above the mean. The total percentage is 50% + 47.5% or 97.5%

 

2. Percentage of heights greater than 67.5: ?% 

 

67.5% is the value of the mean + 1 standard deviations. Also note that 1 standard deviations contains 68% of the total area or 34% above the mean and 34% below the mean. We want the total area above 67.5%, so we want the total area that is greater than 1 standard deviation right of the mean. The total area to the right of the mean is 50% and the total are between the mean and +1 standard deviation is 34%, the area to the right of +1 standard deviation is simply 50% -34% or 16%

 

3.Percentage of heights between 60 and 65: ?% 

 

Percentage of heights less than the mean =  50% 

Percentage of heights less than mean - 2 standard deviations = 50% -.47.5% = 2.5%

Percentages of heights between 65% and 60% is just the difference between the two = 50% - 2.5% = 47.5%

 

4. Percentage of heights between 60 and 67.5:?

 

Percentage of heights less than mean 1 standard deviations = 50% + 34% = 84%

Percentage of heights less than mean - 2 standard deviations = 50% -.47.5% = 2.5%

Percentages of heights between 67.5% and 60% is just the difference between the two = 84% - 2.5% = 81.5%

 

Hope this helps