13.9: Formula Review

Statistics for Public Health (HLTH 10) · LibreTexts

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    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

    Test of a Single Variance

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

    • \(n\) : sample sizes: 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.

    Goodness-of-Fit Test

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

    • O: observed values
    • \(E\) : expected values
    • \(k\) : number of different data cells or categories
    • \(d f=k-1\) degrees of freedom

    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 \(E=\dfrac{\text { (row total)(column total) }}{\text { total surveyed }}\).

    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

    13.9: Formula Review is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by LibreTexts.