Question

The mean sat verbal score is 486, with a standard deviation of 95. use the empirical rule to determine what percent of the scores lie between 391 and 486. (assume the data set has a bell-shaped distribution.)

Answer 1

Find the z-scores for the two scores in the given interval.

$z=\frac{x-\mu}{\sigma}$

For the score x =391, $z=\frac{391-486}{95}=\frac{-95}{95}=-1$.

For the score x = 486, $z=\frac{486-486}{95}=0$

Now you want the area (proportion of data) under the normal distribution from z = -1 to z = 0. The Empirical Rule says that 68% of the data falls between z = -1 to z = 1. But the curve is symmetrical around the vertical axis at z = 0, so the answer you want is HALF of 68%.