One factor completely randomized design - 1 way ANOVA
o Recall the data
  - Small beads: 100, 93, 140
  - Medium beads: 105, 45, 85
  - Large beads: 37, 35, 40
o Surmise model
  - recall eit are N(0, s2)....need this for testing but not estimation
  - ti is a fixed but unknow treatment EFFECT
o Average bead stringing time.. can we estimate this from our data?
  - criteria SSeit2  is smallest
  - via calculus
  - <m, t1, t2 , t3 > = <75.55, 35.44, 2.77, -38.22> or <100, 11, -21.67, -62.67>
  - not uniquely defined
o  What can we estimate?
  - functions of <m, t1, t2 , t3 > , the parameters, of a linear model are estimable iff you can
   write it as an expected value of a linear combination of the response variables Yit
  - Sci = 0 is estimable...contrast
o The equations formed are called Normal equations
  - add another constraint
  - how good are these estimates? Gauss-Markov Theorem: These least squares
    estimators of aan estimable function of parameters are unique and the best linear
    unbiased estimators (BLUE)
o Doing tests
  - __need distributional information
  - Yi.  ~ N(m + ti , s2/ri )
  - E(Yi.  ) = m + ti      but what of s2/ri  ?
  - ^
    s2 = SSE/(n-v)
  -  SSE/ s2  ~ c 2(df=n-v)  table A5, p. 705
  - next :  how many observations should we have taken?
o test H0: t1 =t2 =t3
   -  means not same...SSE/(n-v) called MSE
  - means same ... SStot (or SSE0 in 1-way) / (n - 1) 
  - compare and if very different  H0 not true
o more details later here is ANOVA table
Source                           df                           SS                            MS 
Treatment              v-1
Error                      n-v
Ttoal                      n-1