11.8: Chapter Formula Review

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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

\(\mu_{\chi}^{2}=d f\) 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

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