Statistical Hypothesis Testing
Dr. Phillip YAM
2016/2017 Spring Semester
Reference: Chapter 7 of Tests of Statistical Hypotheses by Hogg
and Tanis.
Section 7.1 Tests about Proportions
I
A statistical hypothesis test is a formal method of making
decisions,
STAT2006 Tutorial 8 Solution
28th/29th/31th 2017
1. [5.845, 0.845]
2. [179.148, 607.480]
3. [0, 1.905]
4. (a) The mgf of X is (1 t) , so the mgf of Y =
Pn
i=1 Xi
is
n
X
Pn
1
MY (t) = E exp t
Xi = (1 t) i=1 i , t < .
i=1
Since the mgf uniquely
determines
STAT 2006 Tutorial 9
5th/7th April 2017
1. Confidence interval for variances
2
We can construct a 100(1 )% confidence interval for X
based on the chi-square distribution
with n 1 degrees of freedom:
"
#
2
2
(n 1)SX
(n 1)SX
,
2 (n 1) 21 (n 1)
2
2
If U 2
Theory of Statistical Inference
Dr. Phillip YAM
2016/2017 Spring Semester
Reference: Chapter 10: Section 7 of Some Theory by Hogg and
Tanis.
Section 10.7 ASYMPTOTIC DISTRIBUTIONS OF
MAXIMUM LIKELIHOOD ESTIMATORS
I
I
Recall that the maximum likelihood esti
STAT2006 Tutorial 10
11th/12th April 2017
1. Terminologies of Hypothesis Testing
Statistical Hypothesis: A statement about population(s) parameter(s) . There are two
complementary (mutually exclusive) hypotheses in a hypothesis testing problem, namely Nu
STAT 2006 Tutorial 9 Solution
5th/7th April 2017
1. For a sample size of n = 27, we will have df = n 1 = 26 degrees of
freedom. For a 95% confidence interval, we have = 0.05, which gives
2.5% of the area at each end of the chi-square distribution. We find
STAT2006 Assignment 3 Solution
h
i
s
s
1. Note that x z0.05 n , x + z0.05 n is an approximate 90% confidence interval for . Therefore,
[
x , x + ] is an approximate 90% confidence for if and only if = z0.05 sn . Since z0.05 1.64,
s = 36, = 8, we have the
STAT 2006 Assignment 2 Suggested Solution
1. (a)
E[
n
X
2
(Xi X) ] =E[
i=1
=E[
n
X
i=1
n
X
(Xi (X )2 ]
(Xi )2 + (X )2 2(Xi )(X )]
i=1
=E[
n
X
2
(Xi ) 2
n
X
i=1
=E[
n
X
2
2
(Xi ) n(X ) ], Since
n
X
Xi = n
i=1
n
X
(Xi ) = 0
i=1
Pn
V ar(Xi ), E[(X )2 ] = V
STAT 2006 Assignment 4
Due Date: 17:00, 21th April, 2017
1. Let X and Y denotethe tarsus lengths of male and female grackles, respectively. Assume that X is
N (X , 2 X ) and Y is N (Y , 2 Y ). Given that n = 25, x = 33.80, s2x = 4.88, m = 29, y = 31.66, a
STAT2006 Assignment 3
Due Date: 17:00, 27th March, 2017
1. A manufacturer sells a light bulb that has a mean life of 1551 hours with a standard deviation of 36
hours. A new manufacturing process is being tested, and there is an interest in knowing the mea
Applied Linear Regression
Chapter 7: Model Diagnostics and Remedies
STAT 3008 (CUHK)
Ch7: Model Diagnostics and Remedies
1 / 68
.
Agenda
1
Model Diagnostics
Residuals Plot
QQ-plot
Test for Non-linearity
Test for Outliers
Test for Heteroscedasticity
2
Reme
Chapter 7
Model Diagnostics and Remedies
In this chapter, we first introduce methods for regression model diagnostics, that
is, checking whether the regression model is appropriate for the data. In practice,
the data may depart from the model assumptions
Applied Linear Regression
Chapter 4: Testings and Confidence Intervals
STAT 3008 (CUHK)
Ch4: Testings and Confidence Intervals
1 / 79
.
Agenda
1
Introduction
2
3
Review of Probability Theory
Expectation and Variance
The distributions N, t, 2 and F
Distrib
STAT 3008 Tutorial 4-2
Part1
Chapter 4: Testings and Confidence Intervals
Introduction
Review of Probability Theory
Expectation and Variance
The distributions N , t, 2 and F
Distributions of and I
Distributions of Projections and Sum of Squares
The
Chapter 4
Testings and Confidence Intervals
Parameter estimation, which reports the point estimates of unknown parameters,
is usually the first step of statistical inference. By formulating suitable test statistics and deriving the probabilistic propertie
STAT 3008 Tutorial 10
A review
Chapter 3: Estimations in Regression
Simple Linear Regression
Least Squares Estimation
Computing Least Squares Estimates
Geometric Interpretation
Scatter Plot and Linear Relationship
Subject Space Plot and Goodness of
Chapter 3
Estimations in Regression
3.1 Simple linear regression without intercept
Given the response Y and a predictor matrix X, the simple linear regression refers
to the model
Y = 0 + 1 X + e ,
(3.1)
i.i.d.
where e N (0, 2 In ), i.e., cfw_ei i=1,.,n N
. /
, I
/
x
./
,
* Name; Major:
i) (6) Roughly stretch a subject space plot for y and A.
ii) (12) Consider the regression model y = g + 51A + 21" + e, 6 ~ N (0, 02I10). Find the estimates
of all the unknown parameters.
Ffw z #1#!)
[$15.72. I W
o :54; 45E
Chapter 5
Model Selection
5.1 Introduction
In practical research problems, while we may usually have a variable of interest
Y in mind, sometimes we may not know which predictors X should be included.
Therefore, selecting a suitable set of variables X = (X
Chapter 6
Categorical Predictors and ANOVA
6.1 Categorical Predictors
Recall that a variable can be classified by its structure as a quantitative variable or
a categorical variable:
Quantitative variable is measured on a numeric or quantitative scale, an
Review, Topics on Multivariate Random
Variables, and Beyond
Dr Phillip YAM
2016/2017 Spring Semester
Reference: Chapters 1 to 5 and Chapter 10 of Probability and
Statistical Inference by Hogg and Tanis.
Preface
After taking STAT2001, you are supposed to b
STAT 2006 Assignment 2
Due Time and Date: 5 p.m., 6th March, 2017
1. Let X1 ,P
X2 , . . . , Xn be randomP
samples from Normal distribution with mean and variance
X = n1 ni=1 Xi and S 2 = n 1 1 ni=1 (Xi X).
P
P
(a) Show that E[ ni=1 (Xi X)2 ] = ni=1 V ar(X
STAT2004
SAS for Data Management
Tutorial 2
1
SAS data sets
1.1
Type of data set
Temporary SAS data set: SAS data set that could only be used within the interactive session and will
be deleted when the session terminates.
Permanent SAS data set: SAS dat
STAT2004
SAS for Data Management
Tutorial 5
1
Loop
The loop statement can be used to execute statements with a changing variable many times. Therefore, the
repeated statements can be simplified.
1.1
DO - END statement
DO index = begin TO end [BY incremen
STAT2004
SAS for Data Management
Tutorial 3
1
Assignment statement
Assignment statement is the most fundamental statement in programming. It is going to save the value of
an expression into a variable.
variable=expression;
to assign the evaluated value o