Statistics 24500, Winter 2007
Solutions to Homework 3
1. (Cauchy distribution)
P
X
c
Y
X
X
c and Y > 0 + P
c and Y < 0
Y
Y
X
X
=P
c and Y > 0 + P
c and Y < 0 ( X X and X Y )
Y
Y
X
X
=P
c and Y > 0 + P
c and Y < 0
Y
Y
X
=P
c
where c is some constant
Stat 245 HW1 Solutions
1. If (X np)/ np(1 p) is a pivotal, then for a xed sample size n, that random variable
has the same distribution regardless of the value of the parameter p. In particular, the span
of the support of these distributions should be the
Statistics 245: Homework 2 due Jan 27
When solving the problems below as well as future homework problems, give detailed
derivations and arguments in order to receive credit. In your solution do not forget to
include your name and the homework number. Ple
Statistics 24500
Homework 2 Solutions
Date: April 18, 2013
1. We call the observations from the Microbiological Method and Hydroxylamine Method as (X1 , Y1 ), (X2 , Y2 ), . . . , (X15 , Y15 ).
sD
Var(X Y ) = Var(D1 ) . So sX Y = n and therefore we have X
Homework 5 (due Feb 17)
In your solution include your name and the homework number. Please staple your
pages together. When solving the problems below, give detailed derivations in order to
receive credit.
1. (Maximum likelihood in one-way ANOVA) Consider
Stat 245 HW4 Solutions
1. (Problem 10, p. 506, 3rd Ed.)
The margin of error obtained through the Tukey method is:
sp
meT ukey = qI,I (J 1) () ,
J
while the uncorrected pooled two-sample procedure yields
me = sp tI (J 1) (/2)
1
1
+
J
J
1/2
=
2
sp t
(/2)
J
Statistics 24500, Winter 2006
Solutions to Homework 6
1. (Maximum likelihood in one-way ANOVA) (For notational simplicity, we assume J = Ji in this
solution, but all results also hold in the case of unequal Ji ; compare Rice p. 449 bottom.)
(a) The likeli
Course Outline for Statistics 24500
Statistical Theory and Methods II
Spring 2013
Instructor: Michael Stein
Required text: J. Rice, Mathematical Statistics and Data Analysis , THIRD edition.
The plan is to cover much of chapters 6, 10, 11, 12 and 14 of Ri
Stat 245 Homework 3 Solution
Winter 2011
Note to graders: Please weight each of the 5 problems out of 20 points for a total of 100 points.
1. (t-test as likelihood ratio test)
The case when 2 is known is done in the book, and it is not equivalent to a t-t
Statistics 245 Midterm Exam
Spring 2011
A single handwritten formula sheet (which you must turn in) and a calculator may
be used.
Page one of three pages. In addition, there is a handout giving some data and R
output.
There are 2 questions, each with m
Statistics 24500
Statistical Theory and Methods II.
Course Description: This is the second quarter of a two-quarter sequence. Enrollment in the second quarter alone is permitted, although not recommended. The rst
quarter covered the basics tools from prob
Statistics 24500, Winter 2007
Solutions to Homework 8
You were not required to turn this in.
1. (Prediction interval)
Problem 11 (2nd ed.), 13 (3rd ed.)
(a) Recall from the class the following formulas.
x2
1
+
)
n SSX
1
V ar(1 ) = 2
SSX
x
Cov (0 , 1 ) =
Stat 245 Homework 7
Winter 2011
Additional Question
Suppose X1 , . . . , Xn are i.i.d. N (, 1). Under H0 we assume = 0 and under H1 we assume = 1.
Fix = 0.05. Compute the power of both the Z -test and the sign-test:
Z -test:
The Z statistic is given by Z
#STAT 24500 Homework 6
#Question 1:
sw=function(g)cfw_
ss=0
for (i in 1:length(g)cfw_
m=mean(g)
ss=ss+(g[i]-m)^2
return(ss)
#part a:
pa=c()
for (i in 1:20)cfw_
s=rt(30,1)
g1=s[1:10]
g2=s[11:20]
g3=s[21:30]
m1=mean(g1)
m2=mean(g2)
m3=mean(
Stat 245 Homework 2
April 26, 2016
1. Let Y be a standard normal variable such that X =Y 2 X12 .
For x 0, FX (x) = P (X < x) = P (Y 2 < x)
= P ( x < Y < x) = FY ( x) FY ( x) = 2FY ( x) 1
R x 1 t2
x
1
x
d
d
d
1
fX (x) = dx
x 2 e 2
FX (x) = 2 dx
FY ( x)
Statistics 245
Due: May 16, 2013
1. Suppose Z1 , . . . , Zn are iid N (0, 1) and, for k = 1, . . . , n, Xk =
Assignment 6
k
j =1 Zj .
(a) Find the joint distribution of X1 , . . . , Xn .
(b) For 1 k n 1, nd the conditional distribution of Xk+1 given Xk =
Stat 245 Homework 1
April 19, 2016
1. We know that P (L < ) = 1 , and P ( < U ) = 1 .
Call L < A and < U B.
We want to find P (A B) = P (A) P (B|A), so in order to find P (L < < U ), we would need to know
P ( < U | > L).
2. p(1 p)(1.962 ) = n(
x p)2 1.962
Stat 245 Homework 3
May 12, 2016
PI
1. The null for one way ANOVA is that for N (= i=1 Ji ) observation of I normally distributed samples
of size Ji with mean i and variance 2 , 1 = 2 = . = I .
PI PJi
2
1
212 (xij i )2
2 N/2 22
i
i=1
j=1 (xij i )
1
L(x1
[L(X1 , ., Xn ), U (X1 , ., Xn )] is a (1 ) confidence interval for if P (L(X1 , ., Xn ) < < U (X1 , ., Xn )
1
Any BVN can be represented as a linear transformation
of two standard normals
Sampling distribution of a normal population:
[L, U ] is ran
Homework 3 (due Feb 3)
In your solution include your name and the homework number. Please staple your
pages together. When solving the problems below, give detailed derivations in order to
receive credit. Also, you may need to use R for certain problems.
Homework 7 Due March 3
1. (Prediction interval) Do problems 13 and 14 on p. 594 in Rice (2007).
2. (Dummy variables) Refer to the data from problem 44 on p. 599 in Rice (2007)
(see data les ASTHMA.DAT and CYSTFIBR.DAT on my website). Let y be the response
Homework 6 Due Feb 24
In your solution please include your name and the homework number. Please staple
your pages together. When solving the problems below, give detailed derivations in order
to receive credit.
1. (ANOVA and normality) Consider the one-wa
Statistics 245 Midterm Exam
Spring 2012
If you correctly follow all of the instructions on this cover page, you will receive 1
bonus point on this exam.
A single handwritten formula sheet (which you must turn in) and a calculator may
be used.
Page one
Statistics 245
Due: June 4, 2013
Assignment 9
Final exam: If you are graduating, your nal is 8:25-10:25 am, Thursday, June 6 in
Eckhart 133. For the rest of you, your nal will be on Tuesday, June 11 from 810 am.
As with the midterm, you should bring a cal
Statistics 245
Due: May 30, 2013
Assignment 8
1. Problem 7, Chapter 14, Rice.
2. Redo parts a, b and d of the previous question for the general regression setting; i.e.,
Y = X + e, where the ei s (the components of e) are independent with mean 0 and
var(e