11.9: Formula Review

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

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