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11.9: Formula Review

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    11.2 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
    \(\mu_{X^2}=d f\) chi-square distribution population mean
    \(\sigma_{\chi^2}=\sqrt{2(d f)}\) Chi-Square distribution population standard deviation

    11.3 Test of a Single Variance

    \(\chi^2=\dfrac{(n-1) s^2}{\sigma_0^2}\) Test of a single variance statistic where:

    \(n:\) sample size
    s: sample standard deviation
    \(\sigma_0\) : hypothesized value of the population standard deviation
    \(d f=n-1\) 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 \(\dfrac{(n-1) s^2}{\sigma_0^2}\), where \(n=\) sample size, \(s^2=\) sample variance, and \(\sigma^2=\) population variance.
    • The test may be left-, right-, or two-tailed.

    11.4 Goodness-of-Fit Test

    \(\sum_k \dfrac{(O-E)^2}{E}\) goodness-of-fit test statistic where:

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

    \(k\) : number of different data cells or categories
    \(d f=k-1\) degrees of freedom

    11.5 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 \(\sum_{i \cdot j} \dfrac{(O-E)^2}{E}\) 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 \(E=\dfrac{\text { (row total)(column total) }}{\text { total surveyed }}\).

    11.6 Test for Homogeneity.

    \(\sum_{i \cdot j} \dfrac{(O-E)^2}{E}\) 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
    \(d f=(i-1)(j-1)\) Degrees of freedom

    11.9: Formula Review is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.