Formula Review

11.1Facts About the Chi-Square Distribution

χ2 = (Z1)2 + (Z2)2 + . . . (Zdf)2 chi-square distribution random variable

μχ2 = df chi-square distribution population mean

$σχ2=2(df)σχ2=2(df)$ chi-square distribution population standard deviation

11.2Goodness-of-Fit Test

$∑k(O−E)2E∑k(O−E)2E$ 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

11.3Test 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 $Σ(i⋅j)(O–E)2EΣ(i⋅j)(O–E)2E$ 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 $E=(row total)(column total)total surveyedE=(row total)(column total)total surveyed$.

11.4Test for Homogeneity

$∑i⋅j(O−E)2E∑i⋅j(O−E)2E$ 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

11.6Test of a Single Variance

$χ2=χ2=$$(n−1)⋅s2σ2(n−1)⋅s2σ2$ Test of a single variance statistic where

n = sample size

s = sample standard deviation

σ = 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 $(n–1)⋅s2σ2(n–1)⋅s2σ2$, where n = the total number of data, s2 = sample variance, and σ2 = population variance.
• The test may be left-, right-, or two-tailed.