1. Chapter 22 Inferences From Nominal Data: the Χ 2 Statistic

2. Χ 2 test of frequencies Pronounced “ki - square” Can be used for nominal level data –data with only the category property The Χ 2 test is based on a comparison of expected frequencies (f e ) versus observed frequencies (f o )

3. Example of frequencies (refresher) Twenty-five miners were asked about their political affiliation - Republican, Democrat, Independent, None Since the data, political affiliation, have only the category property, they are nominal level, and can only be counted

4. Observed Frequencies

5. What was expected? Let’s say, that on average, throughout the country, the percentages of people who report the following political party affiliations are: Democrat - 45% Republican - 40% Independent - 10% None - 5%

6. Is the survey of miners different from the national average? Computing expected frequencies is then done by multiplying the number of observations with the percent expected: f e = n(% expected)

7. Expected frequencies On average, then we would expect: Democrats = 25 (.45) = 11.25 Republicans = 25 (.40) = 10 Independents = 25 (.10) = 2.5 None = 25 (.05) = 1.25

8. Χ 2 statistic Χ 2 statistic is calculated using the following formula:

9. Χ 2 –test on miners political affiliation Partyfofo fefe (f o – f e )(f o – f e ) 2 f e Dem611.25-5.2527.56252.45 Rep10 000 Ind52.5 6.252.5 None41.252.757.56256.05

10. df in Χ 2 test df = number of categories - 1 In the present case, 4 categories - 1 = 3

11. Interpretations of Χ 2 statistic If the differences between the observed and expected frequencies are small, these differences squared will also be small, thus making the sum of these squared differences small also Thus, small differences between observed and expected = small Χ 2 statistic

12. Interpretations of Χ 2 statistic However, if the differences between observed and expected frequencies are large, then these differences squared will be large also, making the sum of the squared differences large Thus, large differences = large Χ 2 statistic

13. Interpretations of Χ 2 statistic How large is large? Fortunately, the Χ 2 statistic is based on a probability distribution of known parameters Table G in your text provides critical values for Χ 2 tests

14. Survey of Miners H O : Frequency of Dems = 11.25 Frequency of Reps = 10 Frequency of Ind = 2.5 Frequency of None = 1.25 H A :H O is incorrect Χ 2 =.05, Χ 2 crit.05 (df = 3) = 7.81

15. Survey of Miners Since the obtained Χ 2 = 11.00 is larger than the critical Χ 2 = 7.81, we Reject H O that the obtained frequencies are 11.25, 10, 2.5, and 1.25 respectively, and Conclude that the obtained frequencies were different than expected