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Inferences From Nominal Data: The Χ2 Statistic

The Χ2 test of frequencies, pronounced "ki-square", can be used for nominal level data with only the category property. It involves comparing expected frequencies (fe) with observed frequencies (fo). This chapter provides an example of using the Χ2 test to analyze political affiliation data from miners and determine if it differs from the national average.

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Inferences From Nominal Data: The Χ2 Statistic

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  1. Chapter 22 Inferences From Nominal Data: the Χ2 Statistic

  2. Χ2 test of frequencies • Pronounced “ki - square” • Can be used for nominal level data • data with only the category property • The Χ2 test is based on a comparison of expected frequencies (fe) versus observed frequencies (fo)

  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: fe = 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. Χ2 statistic • Χ2 statistic is calculated using the following formula:

  9. Χ2 –test on miners political affiliation

  10. df in Χ2 test • df = number of categories - 1 • In the present case, 4 categories - 1 = 3

  11. Interpretations of Χ2 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 Χ2 statistic

  12. Interpretations of Χ2 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 Χ2statistic

  13. Interpretations of Χ2 statistic • How large is large? • Fortunately, the Χ2 statistic is based on a probability distribution of known parameters • Table G in your text provides critical values for Χ2 tests

  14. Survey of Miners HO : Frequency of Dems = 11.25 Frequency of Reps = 10 Frequency of Ind = 2.5 Frequency of None = 1.25 HA: HO is incorrect Χ2 = .05, Χ2crit .05 (df = 3) = 7.81

  15. Survey of Miners • Since the obtained Χ2 = 11.00 is larger than the critical Χ2 = 7.81, we • Reject HO that the obtained frequencies are 11.25, 10, 2.5, and 1.25 respectively, and • Conclude that the obtained frequencies were different than expected

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