8.6 Carrying Out a Chi-Square Test for Homogeneity or Independence

Now that we have chosen the correct test, checked our necessary conditions, and written our hypotheses for our test, it is now time to actually carry out the test! As with our GOF test, the test will consist of two mathematical elements:  the test statistic (χ2 statistic) and our p-value.

The first thing we need to calculate in order to finish our test is our χ2 value which is found using the formula on the formula sheet given for the exam. A much easier way of finding the test statistic is to use our graphing calculator.

Our degrees of freedom are found by taking the number of rows and subtracting 1 and multiplying by the number of columns minus 1.

(number of rows - 1)(number of columns - 1)

Once you finally get your χ2 value, you calculate your p-value by finding the probability of getting that particular χ2 by random chance. As always, if our p is low, we reject the Ho. 

As mentioned above, the best way of doing all of this together is using your graphing calculator device and performing the χ2 GOF test. Just be sure to write out your χ2 value and your p-value from your calculator output. 

Just as we concluded hypothesis tests in previous units, we must compare our p-value from our calculator to a given ɑ value. If it is less than our alpha, we conclude that we reject the Ho and have convincing evidence of the Ha. Otherwise, we fail to reject the null and do not have convincing evidence of the Ha. Remember two things:

Since our p-value(~0) is less than 0.05, we reject the null hypothesis. We have convincing evidence that at least one of the proportions for how people rank on the happiness scale is incorrect.

Template:  Since our (p value) is </> 0.05, we reject/fail to reject our null. 

Independence:  We have/do not have convincing evidence that there is an association between variable x and y in our intended population

Or

Homogeneity:  We have/do not have convincing evidence that the distribution of categorical variable x is different between population x and population y. 🎥  Watch: AP Stats Unit 8 - Chi Squared Tests