424 Stat discussion #3
TA Oce & Phone E-mail Oce hour Tao Yu 1275A MSC , 262-1577 [e-mail] 1:00-2:00pm Tue. and 1:00-2:00pm Wed.
Summary
I. Multiple Regression: Refer to lecture notes Unit1 II. Experiments with a single factor 1. Model : Yij = i + eij or Yij = + i + eij . 2. Assumptions (a) Observations within a treatment and across all treatments are independent
2 (b) Normal assumption:Yij N(i , i )
i = 1, . . . , a, = ... =
2 a
j = 1, . . . , ni
(c) Equal variance 3. Estimates: i = yi ,
2 assumption:1
=
2 2
ij = yij yi HA :not all i are equal
4. Hypothesis H0 : 1 = 2 = ... = a 5. Notations
a ni
a
ni
SST =
i=1 j=1 a
(xij x )2 =
i=1 j=1 a 2
(xij )2
(x )2 N
SStrt =
i=1 a
ni ( x ) = x
i=1 ni
(x )2 1 2 (xi ) ni N
a
SSE =
i=1 j=1
(xij xi ) =
i=1
2
(ni 1)s2 i
6. Relationship: SST = SSE + SStrt 7. ANOVA table Source Treatment Error Total where N =
a i=1
df a1 N a N 1
SS SStrt SSE SST
MS MStrt =SStrt /(a 1) MSE = SSE /(N a)
ni
M Strt M SE
8. Test statistics : F =
1
9. P-value : P (F Fa1,N a ) 10. Properties of the F-test: If all the assumptions the for model are satised, say, 2(a),(b),(c) are all satised, given any value of (type I error), the F-test has the smallest probability of a type II error. The F-test is robust against the violations of the normality and constant variance but not robust against the leak of independence.
Examples
1. Hints for hwk2 question 3 and 4. 2. An experiment was performed to study the eect of polypropylene bers on the compressive strength of concrete. Fifteen concrete cubes were made and randomly assigned to ve levels of ber content (0, 0.25, 0.50, 0.75 and 1%), with three cubes to each level. The mean and standard deviations of the strength data are shown in Tabl...