STAT505 Homework #4
1. J&W Exercise 4.21.
(a) X has a four-variate normal distribution with mean and covariance matrix
(1/60).
(b) (X1 ) 1 (X1 ) is chi-square distributed with p = 4 degrees of freedom.
(c) n(X ) 1 (X ) is chi-square distributed with p = 4
STAT505 Homework #1
1. (taken from J&W Exercise 1.3 )
The following are ve measurements on the variables x1 , x2 , and x3 :
x1
x2
x3
9265
12 8 6 4
3402
8
10
1
Find the arrays x, Sn , and R
The rst sample mean is found by averaging over the ve observations
Problem 1 (adapted from J&W Exercise 5.19 )
Measurements of x1 = stiness and x2 = bending strength for a sample of n = 30 pieces of a
particular grade of lumber are given in the data set lumber.dat (the rst column is x1 ). The
units are pounds/(inches)2 .
STAT505 Homework #2
1. (taken from J&W Exercise 3.14 )
Consider the data matrix
X1 X2
9
1
.
5
3
1
2
We have n = 3 observations on p = 2 variables X1 and X2 . Dene the vector X = [X1 X2 ],
and form the linear combinations
]
[
X1
= X1 + 2X2
c X = [1 2]
X2
b
STAT505 Homework #3
1. J&W Exercise 4.3
Since multivariate random variables are independent if and only if their covariance is
zero, we need only to check the covariance between them.
(a) X1 and X2 are not independent since their covariance is 2.
(b) X1 a
STAT505 Homework #6
1. J&W Exercise 6.19, parts a and c. The le milk.dat contains the data with columns fuel,
repair, capital, and truck, respectively.
(a) The null hypothesis is H0 : 1 = 2 , where 1 is the mean cost vector for gasoline trucks,
and 2 is t
1. (a) From Homework 6, we found significant dierences among the varieties for variables X2
and X3 , so we would expect them to be eective discriminators in this setting.
(b) Let pi denote the prior probability of membership to variety i (before any data
1. (a) The null hypothesis is H0 : 1 = 2 , where 1 is the mean cost vector for gasoline trucks,
and 2 is the mean cost vector for diesel trucks. The alternative hypothesis, Ha , is that
these vectors dier in at least one of their three components: fuel, r
STAT505 Homework #8
In this study, eight people were given drug A, and another eight people were given drug B. Each
persons heart rate was measured at four time points (after 5, 10, 15, and 20 minutes) from when
the drug was given. The data are available
STAT 505 Fall 2016
Homework #2 Solution
1. In multivariate normal distribution, two normal random variables are independent if and only if their
covariance is zero.; if the random variables are not normal, statistical independence implies zero
covariance
(a) The null hypothesis is H0 : no interaction between time and drug. The degrees of freedom
for the hypothesis and error are 3 and 42, respectively. The p-value is less than .0001, which
is signicant evidence to reject H0 and say that the eect of drug is
1. (a) The MANOVA null hypothesis of interest is H0 : 1 = 2 = 3 , where i is the vector
of mean widths for the ith iris type. The sum of squares and cross products (SSCP)
matrices for this hypothesis and error, respectively, are
[
]
[
]
11.3449 22.9327
16
1. Twenty drivers participate in a repeated measures experiment done to investigate how talking
on a cell phone aects driving skill. Each driver responds to driving situations on a simulator
while experiencing three dierent conditions. The first condition
STAT 505 Homework #10
1. A data set includes six variables that measure the wellbeing of patients undergoing radiotherapy. The variables are symptoms (number of side eect symptoms), activity (amount of daily
activity on a 1-5 scale), sleep (quality of sle
STAT 505 Fall 2016
Homework #3 Solution
1.
a) Xij is the ith pollution measurement for the jth variable. j is the population mean of the
jth variable, and jk is the population covariance between variables j and k. If jk = 0, then
variables j and k are unc
Solution-Homework 2
4.3 Let X be N3 (, ) with 0 = [3, 1, 4] and
1 2 0
= 2 5 0
0
0 2
Which of the following random variables are independent? Explain.
(a) X1 and X2
(b) X2 and X3
(c) (X1 , X2 ) and X3
(d)
X1 +X2
2
and X3
(e) X2 and X2 52 X1 X3
Solution:
STAT505 Homework #7
1. J&W Exercise 6.23. The data can be found in iris.dat. Columns correspond to sepal length,
sepal width, petal length, petal width, and type (1,2,3 for setosa, versicolor, and virginica,
respectively). Note that this problem asks for