1.017/1.010 Class 16
Testing Hypotheses about a Single Population
Formulating Hypothesis Testing Problems
Hypotheses about a random variable
x
are often formulated in terms of its
distributional properties.
Example, if property is
a
:
Null hypothesis
H0:
a
=
a
0
Objective of
hypothesis testing
is to decide whether or not to
reject
this
hypothesis.
Decision is based on estimator
a
of
a
:
ˆ
Reject
H0:
If observed estimate
lies in
rejection region
R
a
ˆ
R
a
0
(
)
Do not reject
H0:
O
therwise (
0
ˆ
a
R
a
∈
0
ˆ
a
a
∉
)
Select rejection region to obtain desired error properties:
Test Result
Do not reject
H0
0
ˆ
a
R
a
∉
Reject
H0
0
ˆ
a
R
a
∈
H0
true
P
(H0H0) =1 
α
P
(~H0H0) =
(
Type
I
Error
)
True situation
H0
false
P
(H0~H0) =
β
(
Type
II
Error
)
P
(~H0~H0) =
1
Type
I
error probability
α
is called the test
significance level
.
Deriving Hypothesis Rejection Regions for Large Sample Tests
Hypothesis test is often based on a
standardized statistic
that depends
on unknown true property and its estimate.
Basic concepts are the same
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 Spring '05
 GeorgeKocur
 Statistics, Null hypothesis, Statistical hypothesis testing, Statistical significance, Type I and type II errors, rejection region

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