Chi Squared (X2) Test For Independence
Chi Squared (X2) Test For Independence (Using the TI-83 or 84 Calculator)
The thing we’re looking for: Whether or not 2 variables are independent from each other (don’t rely on the other’s outcome)
Begin w/ the matrix of the observed data
Ex. A Football Team’s Scoring Record
| |Home |Away |Totals |
|Won |97 |69 |166 |
|Lost |42 |83 |125 |
|Total |139 |152 |291 |
This chart is called a
contingency table (has
the original information)
and is also a 2x2 chart
(will be used later)
First is the MANUAL way to find the expected values of each data point and make the expected data table, as well as finding X2 calc. (the basic value of X2)
1. To fill the second identical table with the expected data points, look at the position of the data point and do the row total times the column total, then divide it by the overall total. So the first box has 97. The row total is 166, and the column total is 139, so you multiply 166 and 139, then divide by 291, the overall total. Your answer, 79.29, is the expected total that goes in that space.
| |Home |Away |Total |
|Won |79.29 |86.71 |166 |
|Lost |59.71 |65.29 |125 |
|Total |139 |152 |291 |
This chart is called the expected
frequency table
X2 Test Statistic
Let fo be the observed freq. for each box
Let fe be the expected freq. for each box
X2 calc= (the sum of the numbers you get from each box with this equation) [(fo-fe)2]
Use this equation to find the number for each box (see below), and then add them all fe
together to get X2 calc.
| |Home |Away |
|Win |3.96 |3.62 |
|Lost |4.80 |5.25 |
Here’s the box for those values, and X2 = 17.6253
Second is the EASY CALCULATOR way to find the expected value table as well as X2
2nd ( Matrix ( Edit ( Which ever list you want it in
Enter the dimensions and the numbers of the observed data (first 2x2, then each cell #)
After that’s done, go to the home page
Stat ( Tests ( C: X2 – Test ( Calculate
Then it will display X2 as well as p, which I’ll explain later.
*This step has also created the expected freq. table for you, so to see it, just go to stat ( matrix( edit, and choose the list that comes after the one you inputted your information into.
Now here is what X2 is used for
For your data, you have to state something called the Null Hypothesis, which always says that the two variables you are comparing are independent from each other. The Alternative Hypothesis says that the 2 variables aren’t independent.
There are 2 outcomes to the chi squared test:
1. You reject the null hypothesis, or
2. You don’t reject it (which means they are independent)
Calculate the “critical value” of chi squared to choose one of the above.
Don’t worry, about accepting and rejecting anything until you’ve found X2 calc and X2 crit. (and potentially “p”, see below).
Here’s how to find X2 critical
- First you need the significance level (a %), which should be given in the question. It’s usually 1, 5, or 10 %.
-Subtract the sig. level from 100% (so now the number you’ll use in the football example is 0.95, because 100%-5% is 95%= 0.95)
- The second thing you’ll need is the degree of freedom, which is really easy to find
Degree of freedom = (# of rows-1) x (# of columns-1)
You take these two things and use the special chart, the degree of freedom running vertically and the significance level result horizontally, to find the number that corresponds with both. THAT IS YOUR X2 critical!!
So, here are the different outcomes:
The 2 hypothesis:
Ho (Null) = _____ & ______ are independent
Ha (Alternative) = ______ & ______ are not independent
1. Accept Ho (Null hypothesis) because X2 calc < X2 crit.
Concluding statement: Therefore ______ & _______ are independent
2. Reject Ho (aka accepting the alternative hypothesis, Ha) because X2 calc > X2 crit.
Concluding statement: Therefore ______ & _______ are not independent
♦There is also ONE other way to find out whether or not to reject the null hypothesis.
If the value of p (found at the same time on the calculator with X2) is smaller than the sig. level (remember, the %), then the null hypothesis can also be rejected. So,
Reject Ho because Sig. level > p
Accept Ho because Sig. level < p
So, the two ways to reject the null hypothesis are if
1.) X2 calc > X2 critical, or
2.) Sig. level > p THE END ( YAY!!!
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