## 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