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
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
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 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
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
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
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
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 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
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
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
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
Homework 4 (due Feb 10)
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. (Normal probabilities) Do Problem 10, p. 506 i
#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 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
Data and the plug-in principle
S520
September 22, 2016
Reference: Trosset chapter 7
Reading data into R
Data as a vector
You can enter data manually using the c() function.
die = c(1, 2, 3, 4, 5, 6)
However, with more than a few data points, youll want to
Two sample inference
S520
October 25, 2016
Reminder: Questions to ask
1. What is the experimental unit? (The experimental units must be independent.)
2. From how many populations were the experimental units
sampled? (Remember that the units within each po
Lots of data
S520
October 6, 2016
Reference: Trosset chapter 8
The Law of Averages
Lets write an R function to toss coins.
tosser = function(n, p)cfw_
outcomes = c("Heads", "Tails")
return(sample(outcomes, size=n, replace=TRUE, prob=c(p, 1-p)
Try it out
Goodness of fit
S520
November 15, 2016
First: Read Trosset chapter 13.2 for the general set-up for chi-squared tests. There are two ways to do the
chi-squared test: the likelihood ratio version and Pearsons version. If youre going to do more statistics,
I
One sample location problems
S520
October 20, 2016
Questions to ask
Before rushing into doing a significance test or finding a confidence interval, here are some question you
should ask. The answers will help you work out what method you should use.
1. Wh
Testing examples
S520
October 18, 2016
Significance testing: The steps
1. Write down the null hypothesis (H0 ) and the alternative hypothesis (H1 ). Choose a test statistic that
you can find the distribution for when the null hypothesis is true (plus any
Analysis of variance
S520
November 10, 2016
Reference: Trosset chapter 12
Motivation: Special case (equal sample sizes)
Weve done one- and two-sample tests; now lets generalize to tests where we have k independent samples.
(Of course, if k is 1 or 2, well
Lots of data
S520
July 6, 2016
Reference: A bit of Trosset chapter 9 but mostly chapter 8
Motivation
I take a simple random sample of 100 IU Bloomington faculty (as of 2014) and look up their salaries
online. The data is in the file faculty100.txt. We can
Testing examples
S520
October 18, 2016
Significance testing: The steps
1. Write down the null hypothesis (\(H_0\) and the alternative hypothesis (\(H_1\). Choose a test
statistic that you can find the distribution for when the null hypothesis is true (plu
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)
[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