Skip to main content
Statistics LibreTexts

11.9: Chapter Formula Review

  • Page ID
  • Facts About the Chi-Square Distribution

    \(x^{2}=\left(Z_{1}\right)^{2}+\left(Z_{2}\right)^{2}+\ldots\left(Z_{d f}\right)^{2}\) chi-square distribution random variable

    \(\mu_{\chi}^{2}=d f\) chi-square distribution population mean

    \(\sigma_{\chi^{2}}=\sqrt{2(d f)}\) Chi-Square distribution population standard deviation

    Test of a Single Variance

    \(\chi^{2}=\frac{(n-1) s^{2}}{\sigma_{0}^{2}}\) Test of a single variance statistic where:
    \(n\): sample size
    \(s\): sample standard deviation
    \(\sigma_{0}\): hypothesized value of the population standard deviation

    \(df = n – 1\) Degrees of freedom

    Test of a Single Variance

    • Use the test to determine variation.
    • The degrees of freedom is the number of samples – 1.
    • The test statistic is \(\frac{(n-1) s^{2}}{\sigma_{0}^{2}}\), where \(n\) = sample size, \(s^2\) = sample variance, and \(\sigma^2\) = population variance.
    • The test may be left-, right-, or two-tailed.

    Goodness-of-Fit Test

    \(\sum_{k} \frac{(O-E)^{2}}{E}\) goodness-of-fit test statistic where:

    \(O\): observed values
    \(E\): expected values

    \(k\): number of different data cells or categories

    \(df = k − 1\) degrees of freedom

    Test of Independence

    Test of Independence

    • The number of degrees of freedom is equal to (number of columns - 1)(number of rows - 1).
    • The test statistic is \(\sum_{i \cdot j} \frac{(O-E)^{2}}{E}\) where \(O\) = observed values, \(E\) = expected values, \(i\) = the number of rows in the table, and \(j\) = the number of columns in the table.
    • If the null hypothesis is true, the expected number \(E=\frac{(\text { row total })(\text { column total })}{\text { total surveyed }}\).

    Test for Homogeneity

    \(\sum_{i . j} \frac{(O-E)^{2}}{E}\) Homogeneity test statistic where: \(O\) = observed values
    \(E\) = expected values
    \(i\) = number of rows in data contingency table
    \(j\) = number of columns in data contingency table

    \(df = (i −1)(j −1)\) Degrees of freedom