Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Statistical Inference: Confidence Intervals and Hypothesis Testing, Slides of Structural Analysis
An introduction to statistical inference, focusing on the concepts of confidence intervals and hypothesis testing. It covers the estimation of population means with unknown variances, the use of student's t-distribution, and the calculation of confidence intervals. The document also discusses the process of making decisions on properties of a population based on observed samples.
Typology: Slides
1 / 57
Related documents
Partial preview of the text
Download Statistical Inference: Confidence Intervals and Hypothesis Testing and more Slides Structural Analysis in PDF only on Docsity!
Monte Carlo simulation approach-
Docsity.com
2
0 1
2
3 4
5
6
7
8 9
10
(^50) -5^0
1 2
3
4
5 6
7
8 9
10
(^50) -5^0
1 2
3
4
5 6
7
8 9
10
(^50) -5^0
1 2
3
4
5 6
7
8 9
10
(^50) -5^0
1 2
3
4
5 6
7
8 9
10
(^50) -5^0
1 2
3
4
5 6
7
8 9
10
(^50) -
0 1
2
3 4
5
6
7
8 9
10
(^50) -5^0
1 2
3
4
5 6
7
8 9
10
(^50) -5^0
1 2
3
4
5 6
7
8 9
10
(^50) -5^0
1 2
3
4
5 6
7
8 9
10
(^50) -5^0
1 2
3
4
5 6
7
8 9
10
(^50) -5^0
1 2
3
4
5 6
7
8 9
10
(^50) -
f^ t^^
x t ^
Ensemble of inputs
Ensemble of outputs
System
Generate ensemble ofinputs obeying prescribedmodel for
( ) f t
Process ensemble of outputsusing statistical tools and arriveat probabilistic model for
( ) x^ t
f^ t^^
x t ^
t^
t
Docsity.com
3
^ ^
^
1
Method of moments
1
Mean, Variance, Skewness, Kurtosis,...
Method of maximum likelihoodEstimate directly the parameters of the pdfquantities like mode, median, range
n^
k i
k
i
X n^
^
Estimation of paramters
, ...
can be estimated using this method.
Docsity.com
44
^
^
^
^
^
^
^
^
^
^
1
2
Let
be a random variable with PDF
,
mean
, and standard deviation
.
Let
be an iid sequence with common pdf
.
That is,
1,^
,
,^
,^
1
i
X^
X
n i
X
i i^
j
i^
i^
X^
X
X^
P^
x^
p^
x
X^
p^
x
X^
X^
i^
j^
n
X^
Var
X
p^
x^
p^
x^
i
^
^
^
Estimation of mean
^
^
1
2
,^
.
1
is an unbiased estimator
of^
with minimum
variance and the lowest variance is
.
n
i i
n
X^
n
n
n
Docsity.com
5
1
2
2
1
Consider the estimator
is an unbiased estimator of
with variance
Let us consider the case in which
is known.
If^
is Gaussian, it w
n
i i
X
n
n
X
Sampling distribution for the estimator of mean
^
^1
ould mean that
is an iid
sequence of Gaussian random variables and consequently
would also be Gaussian distributed. If^
is not Gaussian, by virtue of central limit theorem, for large
, we may s
n i X
X
n
^
till consider
to be Gaussian.
It may be inferred that
~^
,^
or,
~^
N^
N
n^
n
^
^
^
^
^
Docsity.com
6
Confidence interval estimation
Docsity.com
7
^
^
^
^
Consider
~^
0,1. Consider
to be
a specified probability
e.g.,
N
n
^
^
-^ -^ -^ -^
0
1
2
3
4
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05^0
x
k^ /2^ ^
^
1
/ k
Area in thisregion=1-
/^
1 / /
P k^^1
k n
^
^
^
^
^
^
^
^
^
1 /
1
1 /
1
/ 2 1
/ 2
k^ k
Docsity.com
8
^
^
^
^
/^
1
/
/^
1
/
/^
1
/
observed value of
from the sample.
,^
confidence
interval for the population mean
. Suppose, 1-
P k We sa
k n
P^
k^
k
n^
n
k^
k n^
n
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
y with 95% confidence that the true mean is contained in the interval
k^
k n^
n
^
^
^
^
Docsity.com
9
^
^
/^
1
/2 /
1
/ 1
This should be interpreted as the probability thatthe random interval
contains the population mean
is 1
Remember
is a determinist
P^
k^
k
n^
n
k^
k n^
n
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Remark
^
(^1) /
1
/
ic quantity.
is a point estimate &,^
is a confidence interval
estimate.
n
i i
x n k^
k n^
n
^
^
^
^
^
Docsity.com
10
= x -0.4326-1.66560.12530.2877-1.14651.19091.1892-0.03760.32730.
^
10 1 /^
1 /
1
10 0.
1.96 &
95% confidence interval
=^
, 0. 10
10
0.6068, 0.
i i
x
k^
k
^
^
^
^
^
^
Example
^
The point estimate of mean is 0.0013.With 95% confidence we say that the populationmean is contained in the interval
0.6068, 0.. ^
Docsity.com
11
0
100
200
300
400
500
600
700
800
900
1000
(^3210) x -1 -2 -
n
^
1000 1 /^
1 /
1
-0.
1000 0.
1.96 &
95% confidence interval
=^
,^ 0. 1000
10
0.1065, 0.0.
i x i
k^
k
^
^
^
^
^
^
^
^
n =
Docsity.com
12
0 1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
(^43210) -1 -2 -3 -
^
10000 1 /^
1 /
1
10000 0.
1.96 &
95% confidence interval
=^
,^ 0. 10000
10000
-0.0163, 0.
i i
x
k^
k
^
^
^
^
^
^
^
n =
Docsity.com
13
^
^
^
^
^
^
^
^
^
^
^
^
^
^
2
2
1
1
2
2
1
2
1
2
2
1
2 2
2
2
2
2 1
1
with X=
1 1 1 1 1 1
2
1 Note:
&
Similarly we can show that Var
n^
n
i^
j
i^
j
n
i i n
i i n
i^
i
i
i
S^
X^
X^
X
n^
n
S^
X^
X
n
X^
X
n
X^
X^
X^
X
n
X^
X^
n
S
S
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Sampling distribution for variance
4
4
2
4
(^31) n
n^
n
^
^
^
^
^
^
^
Docsity.com
14
^
^
2
2
1
2
2
2
2 2
1 1
is an unbiased and
1
consistent estimator of
.
1
/
If population is Gaussian,
and
are Gaussian.
RHS: sum of squares of Gaussian random variable
n
i i n
i i
i
S^
X^
X
n n^
S^
X^
X
n
X^
X
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
2 2
2
2
(^1 )
(^12)
s
such sums have
distributions.
1
is^
distributed with
-^
dof.
The pdf of such a random variable is given by
1
exp
; 2
1
2
2
n
n n^
S^
n u
p u
u^
u
n
^
^
^
^
^
^
^
^
^
^
^
^
Docsity.com
15
^
2
1 2
2
~^
~^
distribution with
-dofs
X^
Y
T^
n
X
T^
Y^
n
X^
N
Y^
n
n
p^
t^
t
n n^
t n
^
^
^
^
^
^
^
^
^
^
^
Student's t - distribution
Docsity.com
16
1
2
Consider the estimator
is an unbiased estimator of
with variance
~ Student t-distribution with
-1 dofs.
n
i i
X
n
n
n
s^
n s
Sampling distribution for the estimator of meanwith variance not known
^
^
^
^
1
1 2
2
estimate of standard deviation from the sample.
dof
f
f
p^
f
f^
f
f
^
^
^
^
^
^
^
^
^
^
^
Docsity.com
17
^
^
^
^
1
,^1
,^1
2
2
1
1 2
2
Consider the statement
From this one can obtain the confidence intervalfor the population mean.
n^
n
f
f
p^
f
f^
f
P^
s^
n
^
^ ^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Docsity.com
18
0 10
20 30
40
50 60
70
80 90
100
(^3210) -1 -2 -
sample
x
Point estimatesˆ^
ˆ
=-0.0677;
=0.
^
^
^
^
^
100
1
,^
,
-0.
-0.
-0.
-0.
-0.
-0.
-0.
-0.
-0.
-0.
-0.
^
^
^
^
-0.
Example
Docsity.com
- -1.
- -1.
- -0.
- -0.
- -0.
- -0.
- -0.
- -1.
- -0.
- -0.
- -0.
- -0.
- -0.
- -0.
- -0.
- -0.
- -1.
- -0.
- -0.
- -0.
- -1.
- -1.
- -1.
- -0.
- -1.
- -0.
- -1.
- -0.
- -0.
- -0.
- -2.
- -0.
- -1.
- -0.
- -2.
- -0.
- -0.
- -0.
- -1.
- -0.
- -1.
- -0.
- -1. - 0. - 0. - -0. - -0. - -0. - 0. - -0. - -0. - -1. - 0. - 2. - 0. - -1. - 0. - -1. - 0. - -1. - 0. - n = Data used in the example
20
0
10
20
30
40
50
60
70
80
90
100
0.06 0.05 0.04 0.03 0.02 0.01^0 -0.01-0.02-0.03-0.
100*(1-alpha)
mu
lower conf-limitupper conf-limitpoint estimate
5000 n^
Docsity.com
21
10
20
30
40
50
60
70
80
90
100
1.06 1.05 1.04 1.03 1.02 1.01^1 0.99 0.98
100*(1-alpha)
sigma
lower conf-limitupper conf-limitpoint estimate
5000 n^
Docsity.com
22
Factors influencing confidence interval^ The statistic used as the estimate The observations made Confidence level Sampling distribution Sample size
Docsity.com
23
^
^
1 /2^
1
/2
Consider the estimator for population meanwith known variance.
~^
,^
or,
~^
n
i i
X^
N^
N
n^
n^
n
P^
k^
k
n^
n
^
^
^
^
^
^
^
^
^
^
^
^
^
Number of samples needed for a given widthof confidence interval
^
1
/2^1
/2
2
Let
half width of confidence interval
be specified. Minimum number of samples required
w k^1
n
n^
k
w
^
^
^
Docsity.com
24
0
0.1^
0.2^
0.3^
0.4^
0.5^
0.6^
0.7^
0.8^
0.9^
1
(^43210) -1 -2 -3 -4
w
Mean^0
0.1^
0.2^
0.3^
0.4^
0.5^
0.6^
0.7^
0.8^
0.9^
5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 1
MeanLower Conf limitUpper Conf limit
10 log
n
^
1
0.95
Docsity.com
25
0.01^
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
(^510410310210)
half conf width
n
MU-Mw
n
Docsity.com