Stat120C
Homework 1 (due Thursday April 10, 2008)
Instructor: Zhaoxia Yu TA: Vinh Nyugen
Note: Where appropriate, all relevant software code and output should be also handed in. Problem 1. (11.10 of Rice) Verify that the two-sample t test at level
STAT120C: Two-Sample T-test
1
the same variance
Assume that X1 , ., Xm is a sample drawn from N (X , 2 ), and Y1 , ., Yn is a sample drawn from
N (Y , 2 ). Also assume that the two samples are independent. We can summerize these assumptions
using the foll
Theorem A Under the assumptions of the one-way ANOVA model,
E(SSW ) = I(J 1) 2
E(SSB) = J
I
X
2i + (I 1) 2
i=1
Proof:
SSW
J
I X
X
(Yij Yi )2
=
i=1 j=1
I
X
(J 1)Si2
=
i=1
= (J 1)
I
X
Si2
i=1
Because the sample variance Si2 is an unbiased estimator for 2 ,
STAT120C: Analysis of Variance (ANOVA)
So far we have considered only one or two samples. For one sample, we were concerned
by the population mean. For two samples, we were concerned by the difference of two
population means. What if there are more than t
2
Two-way ANOVA
In the one-way design there is only one factor. What if there are several factors? Often, we
are interested to know the simultaneous effects of multiple factors, e.g, gender and smoking
on hypertension. The statistical approach to analyze
Lecture Note: a brief review of one-sample t test and introduction to t test for paired
samples
1. the one-sample t test
The one-sample t test is used to conduct hypothesis testing regarding the mean of a
normal distribution when both the mean and the var
1.3
The F test under unbalanced designs
The test for unbalanced designs is very similar - just replacing J with Ji . In this case,
Ji
I X
X
SSW =
(Yij Yi )2
i=1 j=1
SSB =
I
X
Ji (Yi Y )2
i=1
F =
M SB
SSB/(I 1)
P
=
FI1,Pi (Ji 1)
M SW
SSW/[ i (Ji 1)]
Sourc
. Because the samples are independent, the sample variances Si2 s are independent; therefore, SSW/ 2 2I(J1) .
Note
(1) Part I implies that E(SSW ) = I(J 1) 2 .
(2) Sp2 = M SW =
SSW
I(J1)
is called the pooled sample variance.
Proof of B.2
Consider the samp
Stat120C
Homework 2 (due Thursday April 17, 2008)
Instructor: Zhaoxia Yu TA: Vinh Nyugen
Note: Where appropriate, all relevant software code and output should be also handed in. Prbolem 1. (12.5.2 of Rice) Verify that if I = 2, the estimate s2 of T
Stat120C
Homework 5 ()
Instructor: Zhaoxia Yu
Problem 1 Consider the linear regression model with independent and normally distributed random errors:
yi = 0 + 1 xi + i
where i iid N (0, 2 ), i = 1, 2, , n. Let 0 and 1 denote the the least squares estimate