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Chi-Square Goodness of Fit Test

A chi-square goodness of fit test attempts to answer the following question: Are sample data consistent with a hypothesized distribution?

The test is appropriate when the following conditions are met:

  • The sampling method is simple random sampling .
  • The population is at least 10 times as large as the sample.
  • The variable under study is categorical .
  • The expected value for each level of the variable is at least 5.

Here is how to conduct the test.

  • Define hypotheses. For a chi-square goodness of fit test, the hypotheses take the following form.

    H0: The data are consistent with a specified distribution.
    Ha: The data are not consistent with a specified distribution.

    Typically, the null hypothesis specifies the proportion of observations at each level of the categorical variable. The alternative hypothesis is that at least one of the specified proportions is not true.

  • Specify significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.

  • Find degrees of freedom. The degrees of freedom (DF) is equal to the number of levels (k) of the categorical variable minus one: DF = k - 1 .

  • Compute expected frequency counts. The expected frequency counts at each level of the categorical variable are equal to the sample size times the hypothesized proportion from the null hypothesis

    Ei = npi

    where Ei is the expected frequency count for the ith level of the categorical variable, n is the total sample size, and pi is the hypothesized proportion of observations in level i.

  • Find test statistic. The test statistic is a chi-square random variable (Χ2) defined by the following equation.

    Χ2 = Σ [ (Oi - Ei)2 / Ei ]

    where Oi is the observed frequency count for the ith level of the categorical variable, and Ei is the expected frequency count for the ith level of the categorical variable.

  • Find P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a chi-square, use the Chi-Square Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

See also:   Chi-Square Goodness of Fit Test