Thus we can say
"For 95% of all samples of size n = 25, the interval
X
1.96
n
, X
1.96
will cover the true value of "
n
X
X
1.96
X
0.588
X
1.96
X
n
0.588
n
The 95% value is called the level of confidence. This tells us the probability the interval will
co
STATISTICAL INFERENCE
Estimation of a Population Mean : Z-Confidence Intervals
Suppose we are given the following:
Normal Population: scores on a standardized reading test
Population Mean:
(unknown)
Population St. Dev.:
= 1.5
To estimate we will take a sr
QUESTIONS:
1. (a) In theory, how many of the above intervals would you expect to cover the true population
mean ( = 10) ?
_
(b) In fact how many actually do?
_
(c) If this simulation were repeated would you always find that exactly 38 of the 40 intervals
MATH 721 ASSIGNMENT #1 SOLUTIONS
OCTOBER 20, 2010
1
Question #1 (a): The sequence ffn gn=1 is dened on [0; 1) by
n 1 no
fn (x) = [0;1) (x) min x 4 ; 2 4 :
We have
2
d (fn ; fn+1 ) =
Z
Z
n
2
2
jfn+1 (x)
0
fn (x)j dx
m 1
X
m 1
X
d (fk ; fk+1 )
2
k
4
k=n
k=n
MATH 721 ASSIGNMENT #1 SOLUTIONS
OCTOBER 2, 2012
Question #1:
Let E be the set of all x 2 [0; 1] whose decimal expansion contains only 4 and
s
7
s.
(1) E is uncountable since it can be put into one-to-one correspondence with
the Cantor set C by mapping th
MATH 721 ASSIGNMENT #2 SOLUTIONS
OCTOBER 30, 2012
Exercise #6: Suppose that E = ([1 Kn ) [ N where Kn is compact and N
n=1
is null. Then Kn 2 L and N 2 L and so E 2 L as well.
Conversely, suppose that E 2 L. For each n 2 N there is, by regularity of
1
Leb
MATH 721 ASSIGNMENT #3 SOLUTIONS
DECEMBER 13, 2010
Question #1: Suppose that
X such that
(0.1)
0<c
(X) = 1 and that f is a measurable function on
jf (x)j
C < 1;
for all x 2 X;
for some positive constants c; C. We claim that
Z
lim kf kLp ( ) = exp
ln jf j
MATH 721 ASSIGNMENT #2 SOLUTIONS
NOVEMBER 10, 2010
Question #1: Suppose in order to derive a contradiction, that there is a nitely
additive isometry invariant positive measure on the power set P R3 of three
3
dimensional Euclidean space such that (Q) = 1
MATH 721 ASSIGNMENT #3 SOLUTIONS
DECEMBER 11, 2012
Exercise #11: Suppose in order to derive a contradiction, that there is a
nitely additive isometry invariant positive measure on the power set P R3 of
3
three dimensional Euclidean space such that (Q) = 1