Chapter 3.2, Problem 3.41E

To determine

To Explain: the point that is one standard deviation below the mean is a normal distribution is always the 16^th percentile and the point that is two standard deviations above the mean is the 97.5^th percentile.

Explanation of Solution

There 68 percent of the data would be between one standard deviation below and above the mean and therefore 35 percent between one standard deviation below the mean and mean itself.

Also knowing that 50 percent of the data lies before he mean so therefore

50% - 34% = 16%

Lies before 1 standard deviation below the mean and therefore this is the 16^th percentile

Graph:

95 percent of the data is between two standard deviation which below the mean. Therefore 5 percent is not between these two values

But the normal density curve is symmetric, then 2.5% lies below two standard deviations below the men and therefore in 97.5% of the data lies below two standard deviations above the mean. It means that 2 standard deviations above the mean is the 97.5^th percentile.