The heights of 18-year-old men are approximately normally distributed with mean 68 inches and standard deviation 3 inches. what is the probability that the height of an 18-year-old man selected at random is between 66 inches and 67 inches?

The probability would be 0.1193.

We compute the z-score for each end of this interval.  For the lower end, 66 inches, 

z = (X-μ)/σ = (66-68)/3 = -2/3 = -0.67

For the upper end, 67 inches, 

z = (X-μ)/σ = (67-68)/3 = -1/3 = -0.33

Using a z-table (http://www.z-table.com) we find the area to the left of (lower than) the lower limit is 0.2514.  The area to the left of (lower than) the upper limit is 0.3707.  To find the area between these, we subtract:

0.3707 - 0.2514 = 0.1193.