Problem Set 3 Due July 30
ECON 139/239
2010 Summer Term II
1. Multiple Choice. Explain your response briefly.
(a) Which of the following regressions suffer from perfect collinearity?
a. wage = 0 + 1 male + 2 collegegrad + 3 nocollegedegree + u
b. wage = 1
1
HW#6.
5.4.3, 5.4.4, 5.4.9,
5.5.1, 5.5.2, 5.5.3, 5.5.4, 5.5.5, 5.5.6,
5.5.11, 5.5.15, 5.5.21, 5.5.22, 5.5.30, 5.5.32, 5.5.41
5.4.3
(a) Z is a standard normal random variable with known variance, thus we simply treat 2.81
and 2.75 as coming from a standa
AMS412.01
Homework 4
Spring 2015
Name: _ ID: _ Signature: _
Instruction: Dear students, this homework is due before class on Thursday, 2/26/2015.
i .i .d .
i .i .d .
2
1. Let X1 , , X n ~ N ( 1 , 12 ) , and Y1 , , Yn ~ N ( 2 , 2 ) be two independent rand
Quiz 4 Solutions
i .i . d .
Y1 , , Yn ~ U [ , 0]
(a) Find the MOME for
(b) Find the MLE for
(c) Are the MOME and MLE unbiased estimators of ?
Solution:
1
(a) f ( y ) = , y 0
0
1
y2 0
1
E (Y ) = y dy = [ ] =
[0 ( )2 ] =
2
2
2
E (Y ) = Y
= Y => = 2Y
2
n
Quiz 1. Solutions
Question. If X ~ N( , 2 ), what is its mgf M X (t ) ?
Solution:
1
( x )2
M X (t ) = E (e ) = exp( xt )
exp(
)dx
2 2
2
tX
=
=
1
2 2 xt + x 2 2 x + 2
exp(
)dx
2 2
2
1
x 2 2( 2t ) x + 2
exp(
)dx
2 2
2
1
1
[ x ( 2 2t )]2
= exp( t + 2t 2 )
P a g e 1
Exercises 2.4
2.4.11
(a) Let C=colorblind, M=male, and F=female
P (C ) = P ( M ) P (C  M ) + P ( F ) P (C  F )
= .45(.06) + .55(.075)
= .027 + .004125
= .031125
(b) P (C  M ) = .06
2.4.12
(a) P ( Daily exercise ) =
111
400
50
P ( Daily exerc
One and Twosample ttests
The R function t.test() can be used to perform both one and two sample ttests on
vectors of data.
The function contains a variety of options and can be called as follows:
> t.test(x, y = NULL, alternative = c("two.sided", "less
Significance Testing Using R
In the following handout words and symbols in bold indicate R functions and words and
symbols in italics indicate entries supplied by the user; underlined words and symbols are
optional entries. Shaded text represents examples
AMS412.01
Spring 2015
Practice Midterm
Name: _ ID: _ Signature: _
Instruction: This is a close book exam except for an 8x11 cheat sheet (double sided). Anyone who cheats in
the exam shall receive a grade of F. Please provide complete solutions for full c
Quiz 6.
1. Let
be a random sample from a normal population N(
(a) Derive the distribution of
). Please
=
(b) Derive the distribution of
(c) Derive the distribution of
, where S is the sample standard deviation
i .i .d .
2. Let X i ~ Bernoulli( p) , i 1, 2
Quiz 2. Solutions
1. Linear transformation : Let X ~ N ( , 2 ) and Y a X b , where a&b
are constants, what is the distribution of Y?
Solution:
M Y (t ) E(e tY ) E[e t ( aX b) ] E(e atX bt ) E(e atX e bt )
e E (e
bt
atX
) e e
bt
at
a 2 2t 2
2
exp[( a b)
AMS412
HomeWork # 3. Solutions
Prof. Wei Zhu
1.
The gunner on a small assault boat fires three missiles at an attacking plane. Each has
a 20% chance of being on target. If two or more of the shells find their mark, the
plane will crash. At the same time,
Inference on two population means (and two population variances)
1. The samples are paired
paired samples ttest
2. The samples are independent independent samples ttest
a)
12 2 2 pooledvariance ttest
2
2
b) 1 2
unpooledvariance ttest
(Note: to check
Power of the test & Likelihood Ratio Test
Power Calculation (Inference on one population mean)
Likelihood Ratio Test (one population mean, normal
population)
Truth
H0
Decision
Ha
Type II error
H0
Type I error
Ha
= P(Type II error) = P(Fail to reject H 0
Other Common Univariate Distributions
Dear students, besides the Normal, Bernoulli and Binomial distributions, the
following distributions are also very important in our studies.
1. Discrete Distributions
1.1. Geometric distribution
This is a discrete wai
AMS412 Lecture Notes #2
Review of Probability (continued)
Probability distributions.
(1)
Binomial distribution
Binomial Experiment:
1) It consists of n trials
2) Each trial results in 1 of 2 possible outcomes, S or F
3) The probability of getting a certai
AMS 412
Professor Wei Zhu
Jan 29th
1. Review of Probability, the Monty Hall Problem
(http:/en.wikipedia.org/wiki/Monty_Hall_problem)
The Monty Hall problem is a probability puzzle loosely based on the American television
game show Let's Make a Deal and na
Point Estimators, MLE & MOME
Point Estimators
Example 1. Let X1, X2, ,
be a random sample from N(
Please find a good point estimator for 1.
).
2.
Solutions. 1.
2.
There are the typical estimators for
and
Both are unbiased estimators.
Property of Point Est
Lecture 10. Inference on one population mean &
the Exact Confidence Interval for
when the population is normal &
is known
Motivation & simple random sample
Eg) We wish to estimate the average height of adult US males
Take a random sample.

Simple rando
AMS412
Lecture 9. CramerRao Lower Bound, Efficient Estimator,
Best Estimator
Unbiased Estimator of , say
( ) when there are many of them.
It could be really difficult for us to compare
Theorem. CramerRao Lower Bound
Let
be a random sample from a popula
Confidence Interval, continued; & related sample size calculations
1. Sample size estimation based on the large sample C.I. for p
From the interval p Z 2
p(1 p)
, p Z 2
n
p(1 p)
n
L lengh of your 100(1 )% CI 2 Z 2
p(1 p)
n
L, , p are given and we are
Inference on One Population Mean
Hypothesis Testing
Scenario 1. When the population is normal, and the population variance is
known
Data : X 1 , X 2 , X n
i .i . d .
~ N ( ,
2
)
H 0 : 0
H 0 : 0
H a : 0
H a : 0
Hypothesis test, for instance:
Example:
1. Scenario 1: Review of Confidence Interval for One Population Mean when the
Population is Normal and the Population Variance is Known
Dear students, before we study the new topic today: CLT and the large sample confidence interval for
population mean an
Hypothesis Test on One Population Mean (continued)
Scenario 1: When the population is normal, and the population variance
known
Scenario 2: Any population (usually not normal), but the sample size is large
(n 30)
Scenario 3. Normal Population, but the pop
Lecture Notes #3
A Quick Review of Probability & Statistics
(1)
Normal Distribution
Q. Who invented the normal distribution?
* Left: Abraham de Moivre (26 May 1667 in VitryleFranois, Champagne,
France 27 November 1754 in London, England) *From the Wikip
tTest Statistics
Overview of Statistical Tests
Assumption: Testing for Normality
The Students tdistribution
Inference about one mean (one sample ttest)
Inference about two means (two sample ttest)
Assumption: Ftest for Variance
Students ttest
 For
Lecture 10. Inference on one population mean &
the Exact Confidence Interval for
when the population is normal &
is known
Motivation & simple random sample
Eg) We wish to estimate the average height of adult US males
Take a random sample.

Simple rando
AMS412
Lecture 9. CramerRao Lower Bound, Efficient Estimator,
Best Estimator
U
nbiased Estimator of , say
^ ^ ^
1 , 2 , 3
It could be really difficult for us to compare
^
Var ( i ) when there are many of them.
Theorem. CramerRao Lower Bound
Let
Y 1 ,Y
Power of the test & Likelihood Ratio Test
Power Calculation (Inference on one population mean)
Likelihood Ratio Test (one population mean, normal population)
Truth
H0
Decision
Ha
Type II error
H0
Type I error
Ha
= P(Type II error) = P(Fail to reject H 0
Inference on One Population Mean
Hypothesis Testing
Scenario 1. When the population is normal, and the population variance is
known
Data : X 1 , X 2 , X n
i .i . d .
~ N ( ,
2
)
H 0 : 0
H 0 : 0
H a : 0
H a : 0
Hypothesis test, for instance:
Example: