full2.gif (11401 bytes)
Area Between Z Scores
(Both Z Scores + or Both Z Scores -)
(A worked Problem:  Procedure explained in more detail in Chapter 7)


Problem:  Suppose that a personality test has:

    Mean = 200
    Standard Deviation = 50

  I.  What is the proportion of people taking this test that score between 210 and 230?

II.  What is the proportion of people taking this test that score between 180 and 175?


I.  The proportion of people taking the test that score between 210 and 230

    A. Using the Z score formula convert both 210 and 230 to Z scores

        zformula.gif (1006 bytes)

            zpos20.gif (1200 bytes)

            zpos60.gif (1186 bytes)

    B.  Next we will use Table Z to find the area between the mean and Z (Col. 2) for .60

ncsame1.gif (1016 bytes)

        The area between the mean and Z for .60 = .2257

    C.  Next we will find the area between the mean and Z (Col. 2) for .20.

ncsame2.gif (1016 bytes)

        The area between the mean and Z for .20 = .0793

    D.  Finally, because both the Z scores are positive
             we subtract the Column 2 areas to find the area between the Z scores.

        Area between = .2257 - .0793 = .1464

II.  The proportion of people taking the test that score between 180 and 175

    A. Using the Z score formula convert both 220 and 260 to Z scores

        zformula.gif (1006 bytes)

            zpos20.gif (1200 bytes)

            zpos60.gif (1186 bytes)

    B.  Next we will use Table Z to find the area between the mean and Z (Col. 2) for -.50

ncsame3.gif (1016 bytes)

        The area between the mean and Z for -.50 = .1915

    C.  Next we will find the area between the mean and Z (Col. 2) for -.40.

ncsame4.gif (1016 bytes)

        The area between the mean and Z for -.40 = .1554

    D.  Finally, because both the Z scores are negative
             we subtract the Column 2 areas to find the area between the Z scores.

        Area between = .1915 - .1544 = .0371

Hint:  Remember that even though the Z scores are negative the areas that you find in
            Table Z are always positive values
. Therefore when you are finding the area between
            two Z scores the result will always be positive.

 

Copyright 2004 by Mark W. Vernoy