# 11.9: Chapter Formula Review


## 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

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