solution: There are two ways to approach finding an expected value, but they are numerically
equivalent and you can use either. The first way is to find the probability of being in the desired
location by chance and then multiplying that value times the total in the table (as we found an
expected value with discrete random variables). The probability of being in the “Black” row is
and the probability of being in the “Do Not Favor” column is . Assumingindependence, the probability of being in “Exp” by chance is then . Thus, .
The second way is to argue, under the assumption that there is no relation between ethnicity and
opinion, that we’d expect each cell in the “Do Not Favor” column to show the same proportion of
outcomes. In this case, each row of the “Do Not Favor” column would contain of the row total. Thus,
. Most of you will probably find using the calculator easier.Calculator Tip: The easiest way to obtain the expected values is to use your calculator. To do this,
let’s use the data from the previous examples:In mathematics, a rectangular array of numbers such as this is called a matrix. Matrix algebra is a
separate field of study, but we are only concerned with using the matrix function on our calculator to
find a set of expected values (which we’ll need to check the conditions for doing a hypothesis test
using the chi-square statistics).
Go to MATRIX EDIT [A] . Note that our data matrix has three rows and two columns, so make the
dimension of the matrix (the numbers right after MATRIX [A] ) read 3×2. The calculator0 expects you
enter the data by rows, so just enter 130, 120, 75, 35, 28, 12 in order and the matrix will be correct.
Now, QUIT the MATRIX menu and go to STAT TESTS χ 2 -Test (Note: Technically we don’t yet know