http://www.ck12.org Chapter 10. Chi-Square
Ei=expected frequency value for each event
Using this formula and the example above, we get the following expected frequencies and Chi-Square calculations.
TABLE10.8:
Democratic
candidateDemocratic
candidateDemocratic
candidateRepublican
candidateRepublican
candidateRepublican
Candidate
Obs. Freq. Exp. Freq. (O−E)^2 /E Obs. Freq. Exp. Freq. (O−E)^2 /E
Female 48 46. 26. 07 28 29. 74. 10
Male 36 37. 74. 08 26 24. 26. 12
Totals 84 54and the Degrees of Freedom= (C− 1 )(R− 1 )
df= ( 2 − 1 )( 2 − 1 ) = 1Using the table and formula above, we see that the Chi-Square statistic is equal to the sum of all of these values for
(O−E)^2 /E.Therefore,
x^2 =. 07 +. 08 +. 10 +. 12 = 0. 37Using an alpha level of .05, we look under the column for.05 and the row for Degrees of Freedom(d f= 1 ). Using
the standard Chi-Square distribution table, we see that the critical value for Chi-Square is 3.84.Therefore we would
reject the null hypothesis if the Chi-Square statistic is greater than 3.84.
RejectH 0 :OifX^2 > 3. 84
Since our calculated Chi-Square value of 0.37 is not greater than 3.84, we fail to reject the null hypothesis. Therefore,
we can conclude that females are not significantly more likely to vote for democratic candidates than males. In other
words, these two factors appear to beindependentof one another.
Test of Homogeneity
The Chi-Square Goodness-of-Fit and Test of Independence are two ways to examine the relationships between
categorical variables. But what test do we use if we are interested in testing whether or not the assignments of
these categorical variables are random? We perform theTest of Homogeneity,which is computed the same way as
the Test of Independence, to examine the randomness of a sample. In other words, the Test of Homogeneity tests
whether samples from populations have the same proportion of observations with a common characteristic.
The Test of Homogeneity is used when we examine the probability that the assignment of one variable is equal
to another. For example, we found in our last Test of Independence that the factors of gender and voting patterns
were independent of one another. However, remember that our original question was if females were more likely to
vote for Democratic candidates when compared to males. We would use the Test of Homogeneity to examine the
probability that choosing a Democratic candidate was the same for females and males.
Another commonly used example of a Test of Homogeneity is comparing dice to see if they all work the same way.
Let’s use that example to conduct a sample Test of Homogeneity.
Example:A manager of a casino has two potentially ’loaded’ (’loaded dice’ are ones that are weighted on one side
so that certain numbers have greater probabilities of showing up) that they want to examine. The manager rolls each
of the dice exactly 20 times and comes up with the following results.