STATISTICAL INFERENCE FOR ONE-WAY ANALYSIS OF VARIANCE

0. Assume that the data

Sample	Sample	...	Sample
from	from		from
population 1	population 2		population I
$x_{11}$	$x_{21}$	$\ldots$	$x_{I1}$
$x_{12}$	$x_{22}$	$\ldots$	$x_{I2}$
$x_{13}$	$x_{23}$	$\ldots$	$x_{I3}$
$\ldots$	$\ldots$	$\ldots$	$\ldots$
$x_{1n_{1}}$	$x_{2n_{2}}$	$\ldots$	$x_{In_{I}}$

are values of independent normal random variables $X_{ij}$ ( $i = 1, \ldots, I$ and $j = 1, \ldots, n_i$) with mean $\mu_i$ and constant standard deviation $\sigma$. Alternatively, each $X_{ij} = \mu_i + \epsilon_{ij}$ where the $\epsilon_{ij}$'s are independent $N(0, \sigma)$ random errors. Let $N = n_1 + n_2 + \ldots + n_I$, the total number of observations.

The parameters of this model are the population means $\mu_1, \mu_2, \ldots, \mu_I$ and the common standard deviation $\sigma$.

1. Test $H_0: \mu_1 = \mu_2 = \ldots = \mu_I$ against $H_a:$ not all of the means $\mu_i$ are equal.

2. The test statistic is the $F$ statistic (named for statistical pioneer Sir Ronald Fisher). This is traditionally displayed in an ANOVA (ANalysis Of Variance) table, even though we need only the $F$ value and its two degrees of freedom.

Source of	Degrees of	Sum of	Mean	F	P
variation	freedom	squares	square	value
Between
groups	$DFG = I-1$	$$SSG = \sum n_i(\overline{x}_i - \overline{x})^2$$	$MSG = SSG/DFG$	$F = MSG/MSE$
(model)
Within
groups	$DFE = N - I$	$$SSE = \sum (n_i - 1)s_i^2$$	$MSE = SSE/DFE$
(error)
Total	$DFT = N-1$	$$SST = \sum_{i,j} (x_{ij} - \overline{x})^2$$	$MST = SST/DFT$

3. Under the model assumptions and the null hypothesis, the $F$ statistic has an $F$ distribution. Critical values of the $F$ statistic are given in Table E of our text. There are two parameters: the numerator degrees of freedom and the denominator degrees of freedom. To do one-way ANOVA on the TI-83 Plus, enter the data into lists and then do this: STAT TESTS F:ANOVA( $L_1, L_2, \ldots, L_I$) ENTER. This will give you a $P$ value, too.

This procedure is called ``analysis of variance'' because, it turns out, SST = SSG + SSE, i.e., the total sum of squares equals (is ``analyzed'' into) the between-groups sum of squares and the within-groups (error) sum of squares.