### 11.1 Facts About the Chi-Square Distribution

*χ*^{2} = (*Z*_{1})^{2} + (*Z*_{2})^{2} + … (*Z _{df}*)

^{2}chi-square distribution random variable

*μ _{χ2}* =

*df*chi-square distribution population mean

${\sigma}_{{\chi}^{2}}\text{=}\sqrt{2\left(df\right)}$ Chi-Square distribution population standard deviation

### 11.2 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

- Use the test to determine variation.
- The degrees of freedom is the number of samples – 1.
- The test statistic is $\frac{(n\u20131){s}^{2}}{{\sigma}_{0}^{2}}$, where
*n*= sample size,*s*^{2}= sample variance, and*σ*^{2}= population variance. - The test may be left-, right-, or two-tailed.

### 11.3 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

### 11.4 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 $E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}$.

### 11.5 Test for Homogeneity

$\sum}_{i\cdot j}\frac{{(O-E)}^{2}}{E$ 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