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In the early 1900's
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Two different resear
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For this research situation, decid
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Question 1
The amount of money college students spend each semester
Quiz #1 (Once you start your quiz, you have 120 min to finish it. Do not logout till you are done.
1/25/13 2:35 PM
KEY Quiz #1
GULEC, CAGATAY
Attempts
1, 2
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STATS 105 HW3 due Thursday in class
Problem 1: Suppose X1 , X2 , ., Xn Bernoulli(p) independently. Find the MLE of p. Suppose
the observed data are 0 0 1 1 1 1 1 0 1 1, compute the MLE.
Problem 2: Suppose X1 , X2 , ., Xn N(, 2 ) independently. Find the ML
STATS 105 HW5 due Thursday in class
Problem 1: Suppose X1 , ., Xn f (x) independently. For X f (x), let = E[X ], and 2 =
Var[X ]. Let
1n
X=
Xi ,
n i=1
and
s2 =
E(s2 )
1
n1
n
(Xi X )2 .
i=1
2.
Prove E[X ] = , and
=
Problem 2: Suppose X1 , X2 , ., Xn N(, 2
STATS 105 HW6 due Thursday in class
Problem 1: Suppose we ip a coin 100 times independently and we get 55 heads.
(1) Test H0 : p = 1/2 vs H1 : p > 1/2, and calculate the p-value.
(2) Suppose the decision rule is to reject H0 if p > .6. What is the probabi
Chapter 9:
9-5
a) = P(reject H0 when H0 is true)
X 11.5 12
/ n 0.5 / 4 = P(Z 2)
= P( X 11.5 when = 12) = P
= 0.02275.
The probability of rejecting the null hypothesis when it is true is 0.02275.
b)
= P(accept H0 when = 11.25) =
=
PX 11.5 | 11.25
X 11
STATS 105 HW2 due Thursday in class
Problem 1: For random variables X , Y and Z , prove
(1) Cov(X, Y ) = E(XY ) E(X )E(Y ), and Var(X ) = E(X 2 ) E(X )2 .
(2) Cov(aX + b, cY + d) = acCov(X, Y ).
(3) Cov(X + Y, Z ) = Cov(X, Z ) + Cov(Y, Z ).
2
2
(4) Let X
STATS 105 HW4 due Thursday in class
Problem 1: Let X Bernoulli(), i.e., P (X = 1) = and P (X = 0) = 1 . Y is continuous.
Given X = 1, the conditional density of Y f1 (y ), and given X = 0, Y f0 (y ).
(1) What is the marginal density of Y ?
(2) Given Y = y
STATS 105 HW1 due Thursday in class
Problem 1: Let X f (x) be a continuous random variable with density f (x). Dene E[h(X )] =
h(x)f (x)dx. Prove
(1) E(aX + b) = aE(X ) + b.
(2) Var(aX + b) = a2 Var(X ).
(3) Let = E(X ) and 2 = Var(X ). Let Z = (X )/ . Ca
STATS 105 Midterm
Problem 1:
(1)
n
L() =
i=1
1
2 2
exp
n
1
l() = log 2 2 2
2
2
(Yi xi )2
2 2
n
(Yi xi )2
i=1
Let l()/ = 0 and l()/ 2 = 0, we obtain
n
i=1 (Yi
xi )xi
=0
2
n
2
n
i=1 (Yi xi )
2 =0
4
2
2
Hence, we have
=
2 =
1
n
n
i=1 Yi xi
n
2
i=1 xi
n
(Y
STATS 105 Homework 7
Problem 1: The decision rule is to reject H0 when
p1
p0
n
i=1
Xi
(1 p1 )n
n
i=1
Xi
n
i=1
Xi
(1 p0 )n
n
i=1
Xi
>C
where C is a threshold.
p1
p0
n
i=1
Xi
n
i=1
p1 1 p0
p0 1 p1
n
i=1
p1 (1 p0 )
p0 (1 p1
n
Xi
log
i=1
n
i=1
n
1 p1
1 p0
Xi
STAT 105 Homework 7 Due Thursday in class
Problem 1: Suppose X1 , ., Xn Bernoulli(p) independently. Suppose we want to test H0 : p = p0
versus H1 : p = p1 (for instance, p0 = .5 and p1 = .7, or p0 = .5 and p1 = .4). Show that if p1 > p0 ,
then the decisio
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If we want to estimate the mean difference in scores on
Describing the Relation Between Two Variables
Learning Objectives
1. Construct and interpret scatter diagrams
2. Compute and interpret correlation coefficients
3. Compute and interpret least square lines
4. Interpret residual plots
Scatter Diagrams; Corre
Summer Session A: Stat 105
Tentative Weekly Plan:
Week
Dates and Material
Week 1
June 24 - June 28: Chapters 6 and 7
Week 2
July 1st - July 5th: Chapter 8
Week 3
July 8 - July 12: Chapters 9 and 10
Week 4
July 15 - July 19: Midterm Exam July 16 and on the
t Table
cum. prob
t .50
t .75
t .80
t .85
t .90
t .95
t .975
t .99
t .995
t .999
t .9995
one-tail
0.50
1.00
0.25
0.50
0.20
0.40
0.15
0.30
0.10
0.20
0.05
0.10
0.025
0.05
0.01
0.02
0.005
0.01
0.001
0.002
0.0005
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.00
Statistical Reasoning for Engineers
University of California Los Angeles (UCLA)
DEPARTMENT OF STATISTICS
COURSE SYLLABUS (TENTATIVE)
Course:
Lecture Meeting:
Quarter:
Professor:
Office:
E-mail:
Phone:
Office Hours:
Prerequisite:
Textbook:
STAT 105: Statis
Estimation of Parameters
The Method of Moments
Notation:
Definition: The kth sample moment is given by:
Suppose we wish to estimate two parameters
moments:
can be expressed in terms of the first two
Then the method of moments estimates are:
Example 1: Poi
Chapter 12:
12-1.
223
553
10
a) X X 223 5200.9 12352
553 12352 31729
1916.0
X y 43550.8
104736.8
b)
171.055
3.713
1.126
c)
y 171.055 3.714(18) 1.126(43) 189.49
so
y 171.055 3.714 x1 1.126 x2
1
12-21.
a) t0
F0
j j0
se( j )
, null hypothesis
S
STATS 105 Homework 6
Problem 1:
(1)
55
= 0.55
100
pobs =
Z=
zobs =
p p0
p0 (1 p0 )/n
0.55 0.5
N (0, 1) under H0
0.5 0.5/100
=1
p -value = P (Z zobs ) = P (Z 1) = 0.158655
(2) Under H0 , p = p0 = .5, p N (p0 , p0 (1 p0 )/n), and
Z=
p p0
p0 (1 p0 )/n
N (0