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This module provides a summary on formulas used in Chi-Square Distribution as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

The chi-square probability distribution

μ = df and σ = 2 df

    Goodness-of-fit hypothesis test

  • Use goodness-of-fit to test whether a data set fits a particular probability distribution.
  • The degrees of freedom are number of cells or categories - 1 .
  • The test statistic is Σ k ( O E ) 2 E , where O = observed values (data), E = expected values (from theory), and k = the number of different data cells or categories.
  • The test is right-tailed.

    Test of independence

  • Use the test of independence to test whether two factors are independent or not.
  • The degrees of freedom are equal to (number of columns - 1)(number of rows - 1) .
  • The test statistic is Σ ( i j ) ( O - E ) 2 E where O = observed values, E = expected values, i = the number of rows in the table, and j = the number of columns in the table.
  • The test is right-tailed.
  • If the null hypothesis is true, the expected number E = (row total)(column total) total surveyed .

    Test of homogeneity

  • Use the test for homogeneity to decide if two populations with unknown distributions have the same distribution as each other.
  • The degrees of freedom are equal to number of columns - 1 .
  • The test statistic is Σ ( i j ) ( O - E ) 2 E where O = observed values, E = expected values, i = the number of rows in the table, and j = the number of columns in the table.
  • The test is right-tailed.
  • If the null hypothesis is true, the expected number E = (row total)(column total) total surveyed .
The expected value for each cell needs to be at least 5 in order to use the Goodness-of-Fit, Independence and Homogeneity tests.

    Test of a single variance

  • Use the test to determine variation.
  • The degrees of freedom are the number of samples - 1.
  • The test statistic is ( n - 1 ) s 2 σ 2 , where n = the total number of data, s 2 = sample variance, and σ 2 = population variance.
  • The test may be left, right, or two-tailed.

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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