Question

A set of data is normally distributed with a mean of 44 and a standard deviation of 3.2. Which of the following statements are not true?

I. 68% of the values are between 37.6 and 50.4.
II. 13.5% is between 37.6 and 40.8.
III. 5% of the values are lower than 37.6 or higher than 50.4.

Answer 1

I) (50.4-44)/3.2=2; (37.6-44)/3.2)=-2. The difference between these is 4 standard deviations which means that by far the majority of the data lies between 37.6 and 50.4 (almost 100%). So the statement is false.

II) The data lies between 1 and 2 standard deviations below the mean (Z-scores between -2 and -1):

0.1587-0.0228=0.1359 or around 13.5%, so the statement is true.

III) This sounds about right: 37.6 is 2 SD's below the mean (2.28%) and 50.5 is 2 SD's above the mean (2.28%), so the relevant combined percentage is 4.56%, close to 5%.