CK-12 Probability and Statistics - Advanced

10.2. Test of Independence http://www.ck12.org

the row and column totals from the table. The expected value for each cell of the table can be calculated using the
formula:

Expected Frequency=

(Row Total)(Column Total)

Total Number of Observations

In the table above, we calculated that the Row Totals are 76 (Females) and 62 (Males) while the Column Totals are
84 (Democrat) and 54 (Republican). Using this formula, we find the following expected frequency for each cell.

Expected Frequency for Female Democratic cell is 76× 84 / 138 = 46. 24

Expected Frequency for Female Republican cell is 76× 54 / 138 = 29. 74

Expected Frequency for Male Democratic cell is 62× 84 / 138 = 37. 74

Expected Frequency for Male Republican cell is 62× 54 / 138 = 24. 26

Using these calculated expected frequencies, we can modify the table above to look something like this:

TABLE10.7:

Democratic Democratic Republican Republican Total

Observed Expected Observed Expected

Female 48 46. 26 28 29. 74 76

Male 36 37. 74 26 24. 26 62

Total 84 54 138

Using these figures above, we are able to calculate the Chi-Square statistic with relative ease.

The Chi-Square Test of Independence

As with the Goodness-of-Fit test described earlier, we use similar steps when running a Test-of-Independence. First,
we need to establish a hypothesis based on our research question. Using our scenario of gender and voting patterns,
our null hypothesis is that there is not a significant difference in the frequencies with which females vote for a
Republican or Democratic candidate when compared with males. Therefore,

Null HypothesisH 0 :O=E(there is no statistically significant difference between observed and expected frequen-
cies)

Alternative HypothesisHa:O 6 =E(there is a statistically significant difference between observed and expected
frequencies)

Using the table above, we can calculate the Degrees of Freedom and the Chi-Square statistic. The formula for
calculating the Chi-Square statistic is the same as before:

χ^2 =∑

i

( (^0) i−Ei)^2
Ei
where:
χ^2 =Chi-Square statistical value
Oi=observed frequency value for each event