36-226 INTRODUCTION TO PROBABILITY & STATISTICS II Midterm II
You must show your work and/or explain your steps in order to get full credit or be considered
for partial credit.
This is a closed book/closed notes exam. You may use a sheet of notes (8.5 by
STAT 226 - HW 9
Due Monday April 24
Do the following problems 1. Let X1 , X2 , . . . , Xn be a random sample from a N (, 2 ) distribution, where 2 is known. The following two simple hypotheses are being tested: H0 : = 0 vs. Ha : = a . where a < 0 . (a) Fi
STAT 226 - Solutions to Homework 7
Problem 1 (9 points: (a) 4 (b) 2 (c) 3)
(a) A 100(1 )% CI for ( is known) is X z/2 . In this case, a 95% C.I. for
n
, the mean ARSMA score of the rst generation Mexican-Americans is: (2.13, 2.59) (i.e.
08
2.36 2 .50 ).
W
STAT 226 - Homework 8
Due Monday April 10
instructions:
1. This homework is due Monday , April 10 in class, BEFORE class starts. No HW
will be accepted after the assignments have been collected.
2. Please remember to staple if you turn in more than one pa
STAT 226 - Solutions to Homework 8
Problem 1
X1 , X2 , ., Xn iid exp( ).
(a)
1
L( ) = ( )n e1/
n
i=1
1
ln L( ) = n ln
ln L( )
n
1
= + 2
xi
n
xi
i=1
n
xi = 0
i=1
n
n +
=
xi = 0
i=1
n
i=1
2 ln L( )
n
2
= 2 3
2
xi
n
n
i=1
=X
n
xi | = X = 2 < 0
X
is the M
STAT 226 - Homework 7
Due Friday March 31
Do the following problems:
1. The Acculturation Rating Scale for Mexican Americans (ARSMA) is a psychological test developed to measure the degree of Mexican/Spanish versus Anglo/English
acculturation of Mexican A
STAT 226 - Solutions to Homework 6
Problem 1
(a)
E(
1
+ n
+X
| ) =
+
E (X | ) = (X | Bin(n, ) =
=0
+n
+n +n
+n
However:
lim
n
(b)
+n
+ +n
=
The prior distribution of , g ( ) is Beta( = 135, = 135).
X | Bin(n = 1000, ), and the data we got is x = 545 po
STAT 226 - Solutions to Homework 4
Problem 1 (20 points: (a) 10 [2 for each expected value] (b) 20 points [2
for each variance, 2 for identifying the one with smallest variance)
Recall that if X exp( ) then:
(i) E(X ) =
(ii) V(X ) = 2
(a)
E(1 ) = E(X1 )
STAT 226 - Solutions to Homework 5
Problem 1
1
n
n n
xi
L( ) = 2
i=1
using the factorization theorem:
h(X1 , ., Xn ) = [
n
i=1
xi ]
g (, T ) = n 2n [
n
i=1
xi ]
1
T =
n
i=1
xi is sucient for
Problem 2
fY (y, ) =
2y
y2
e , y > 0, > 0
n
L( ) =
n n
yi
STAT 226 - Homework 5 - Suciency and MVUE
Due Friday February 24 -BEFORE CLASS STARTS
Do the following problems:
1. Let X1 , X2 , . . . , Xn be a random sample from the following distribution:
2
x>2
x+1
>0
0
fX (x) =
otherwise
n
Show that
Xi is sucient
STAT 226 - Homework 6
Due Friday March 24, before class starts
Topics Covered by this HW assignment:
(i) Bayesian Estimation
(ii) Distributions Derived from the Normal Distribution.
I. Read sections 7.2, 7.3, 8.5, 8.6-8.9
II. Do the following problems:
1.
STAT 226 - Solutions to Homework 3
Problem 1
(14 points: (a) 8 [3 for L(), 3 for ciritical point, 2 for max] (b) 2 (c) 4 [-2 if no
explanation])
a)
n
n
e xi
en i=1 xi
=
L() =
n
xi !
i=1 xi !
i=1
n
n
ln L() = n +
xi ln ln
i=1
xi !
i=1
n
ln L()
xi
= n + i=
36-226 - Solutions to HW 2
Problem 1 15 points: (a) 6 points [3 for E(X), 3 for (V(X)] (b) 5 points (0 if
E(1/X) = 1/E(X), (c) 4 points
a) Recall that if X exp( ) then, E (X ) = and V (X ) = 2 . Therefore,
E (X ) = E (X ) = ,
V (X ) =
b)
n
1
E( ) = E(
X
)
STAT 226 - Homework 4 - Properties of Estimators
Due Friday February 17 -BEFORE CLASS STARTS
I. Read sections 9.4, 8.5, 8.6
II. Do the following problems:
1. Page 368, problem 8.4
Note: It be useful to remember the following result about an the exponentia
STAT 226 - Homework 3 - M.L.E
Due Friday February 10 -BEFORE CLASS STARTS
I. Read sections 8.2, 8.3, 9.2 (if you havent done so already) and 9.4
II. Do the following problems:
1. Page 453 problems 9.72 (a,b,d)
2. Page 453, problem 9.74(b)
3. Page 453, pro
STAT 226 - Homework 2
Due Friday February 3 - BEFORE CLASS STARTS
Topics covered in this assignment:
Examining properties of sample statistics (by reviewing 36-225 material)
Joint density of a random sample (using product rules)
Point estimation using
STAT 226 - Homework 1 - Exploratory Data Analysis
You can learn a lot just from looking at the data!
Due Friday January 27 - Before class starts
Comments:
1. This assignment requires the use of the statistical software Minitab. You can download
Minitab to