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SAMPLE FINAL EXAM
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In an effort to improve customer service, Dell monitors the time customers wait on hold
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HYPOTHESIS TESTING
I. Introduction
Def: A hypothesis test is a statement about a population parameter(s)
Two complementary hypotheses:
H 0 : Null Hypothesis (Simple/Composite)
H1 : Alternative Hypothe
B. Bayesian Tests
-Utilize the posterior distribution ( | x )
-Calculate the P ( 0 | x ) and P ( 0c | x )
-Set up tests to reflect desired probabilities of error
For example:
cfw_x : P( 0c | x) > 1/ 2
II. Types of Hypothesis Tests
A. Likelihood Ratio Tests
The statistic test (x ) is computed as:
(x ) =
sup L ( |x )
0
sup L ( |x )
The supremum in the numerator is relative to the domain of the null
EVALUATING HYPOTHESIS TESTS
I. Best Tests of Hypothesis
C is a best critical region of size for testing simple hypotheses
H 0 : = '
H1 : = '
if for every subset A of the sample space for which P [ ( X
OTHER HYPOTHESIS TESTING ISSUES
II. Asymptotic Results (Ch 10.3)
1) It can often be very difficult to derive the actual distribution of a LRT statistic so that yu
can get the value of c in ( x ) < c .
OTHER HYPOTHESIS TESTING ISSUES
I. Size of U-I and I-U Tests
Recall from our previous lecture:
1). Union Intersection Tests
U-I: H 0 :
True only if All are true.
Specify:
R : cfw_ x : T ( x ) R se
III. Non-UMP Tests
Suppose that UMP tests do not exist.
1. Unbiased Tests
Definition: A test with power function ( ) is unbiased if ( ' ) ( " ) for every
' co , " o .
In common language, the power is
III. Non-UMP Tests (Continued)
2. Locally Most Powerful Tests
Unbiased tests are designed to give high power over all parameter values in the region of
the alternative hypothesis. An alternative is to
METHODS OF EVALUATING ESTIMATORS
There are often multiple point estimators for a parameter(s). Several criteria are used to
compare estimators:
1. Bias: The difference between the expected value of th
Asymptotic variances of MLEs and of functions of MLEs
a) From Casella and Berger, one approach is to use the Rao-Cramer Lower Bound.
h ' ( )
Var h =
.
I n ( )
() )
2
This looks almost like the Rao-
Quartiles
Quartiles are the values that divide a list of numbers into quarters.
First put the list of numbers in order
Then cut the list into four equal parts
The Quartiles are at the "cuts"
Like this
Business Economics Applications
Review of Revenue, Cost and Profit
We define the revenue R to be the total amount of money coming into the company, the
cost C the total amount of money coming out of t
Definition
Mathematics may be defined as the study of relationships among quantities, magnitudes and
properties, and also of the logical operations by which unknown quantities, magnitudes, and
propert
Optimization
Steps for Solving Optimization Problems
1. Draw the picture and label variables.
2. Determine a constraint equation (if necessary) and a maximizing (minimizing)
equation.
3. Use the const
Increasing and Decreasing Functions
Definition of Increasing and Decreasing
We all know that if something is increasing then it is going up and if it is decreasing it is
going down. Another way of say
The First Derivative Test
The First Derivative Test (Motivation and Theorem)
If f is a function, then f has a relative maximum at x = c if for all points a near c, f(c) >
f(a), and f has a relative mi
Higher Derivatives
The Second Derivative
The derivative of the derivative is called the second derivative.
There are two main ways of writing the second derivative. They are
d2y
f '(x)
and
dx2
The mai
Graphing
The Distance Formula
If two points are in the plane, we can find the distance between them by using the
Pythagorean theorem. For example, if the points are (1,4) and (5,2), then the the figur
The Product and Quotient Rules
The Product Rule
Theorem:
Let f and g be differentiable functions. Then
(f(x)g(x)' = f(x)g'(x) + f '(x)g(x)
Proof:
We have
d/dx (fg)
f(x+h) g(x+h) - f(x) g(x)
= lim
Add
II. Corollaries and Extensions to the Neyman Pearson Lemma
A. Recall Neyman Pearson Lemma
Test H0: = o vs H1: = 1 , where X ~ f ( x i ) , i = 0 ,1
(Since two alternatives), using
x R if f ( x | 1 ) >