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 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.
11.4 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
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