68-95-99.7 rule

Question

 Olympic 100 meter dash times (measured in seconds) this year are normally distributed and have a mean time μ = 10.5 and a standard deviation of 𝝈 = 0.5. Use the 68-95-99.7 rule to approximate the percentage of times that are greater than 11 seconds.

Answer 1

Answer: 16%

Explanation:

The empirical (68-95-99.7) rule says that 68% of all values fall within 1 standard deviation of the mean, 95 percent within 2 standard deviations of the mean, and 99.7% within 3 standard deviations of the mean. Therefore, if we consider all values within 3 standard deviations of either side of the mean (positive or negative), we conclude that 99.7%/2 = 49.85% of all values are between -3 standard deviations of the mean and the mean value. If we take the remaining .3%, divide it by 2 (.15%) and add it to the 49.85%, we conclude that 50% of all values are less than or equal to the mean value.

We are interested in all values greater than 11 seconds and 11 seconds is one standard deviation off the mean of 10.5 seconds. Therefore, we are interested in the percentage of values greater than one standard deviation of the mean. Because 68% of all values fall between -1 and 1 standard deviation of the mean, we add 68%/2 = 34% to the 50% to get 84%. The 84% is the percentage of values less than or equal to 11 seconds, so 100%-84% = 16% is the percentage of values greater than 11 seconds.