MAT 2379 - Introduction to Biostatistics
Hypothesis Testing
Professor: Termeh Kousha
Summer 2015
1

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Hypothesis testing, type I and type II er-
rors
Hypothesis testing is a procedure that leads us to decide if experimental data
supports a hypothesis concerning population(s) parameter(s). We will con-
sider hypotheses concerning a population mean
μ
or a population proportion
p
.
Stating the Hypotheses:
Often the researcher would to verify a change
in the unknown parameter under new experimental conditions. For exam-
ple, a manufacturer of a new fiberglass tire claims that the mean life of the
new tires are greater than the mean life of tires using the old manufacturing
process. The previous mean life was 65
,
000 km.
Let
μ
denote the mean life of the new tires.
The no change hypothesis
(that we will call the
null hypothesis
) is
H
0
:
μ
= 65
,
000 and the claim
or research hypothesis (that we will call the
alternative hypothesis
) is
H
1
:
μ >
65
,
000.
We want to test
H
0
:
μ
= 65
,
000
against
H
1
:
μ >
65
,
000
.
Now we consider an example involving a proportion.
Suppose that we
would like to test the hypothesis that the proportion of defective items pro-
duced at a particular plant is
p
= 2%. Then, we would test
H
0
:
p
= 0
.
02
against
H
1
:
p
6
= 0
.
02
.
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Null Hypothesis:
The null hypothesis will always be a simple state-
ment concerning the unknown parameter
θ
. That is, it is a statement of the
form
θ
=
θ
0
, where
θ
0
is some real number. For example,
H
0
:
μ
= 65
,
000 or
H
0
:
p
= 0
.
02. The value of the parameter in the null hypothesis will be the
boundary value of the parameter from the alternative hypothesis.
Alternative Hypothesis:
The alternative hypothesis will be a composite
statement concerning
θ
. It is often the research hypothesis, i.e. the hypothe-
sis that we would like to support with the data. We will consider three types
of alternatives: (
θ
is the unknown parameter and
θ
0
is some real number)
H
1
:
θ < θ
0
is a left-sided alternative;
H
1
:
θ > θ
0
is a right-sided alternative;
H
1
:
θ
6
=
θ
0
is a two-sided alternative
.
Definitions:
•
A
test statistic
is a statistic that is used to test hypotheses.
•
The
critical region
of the test statistic is a set of possible values of
the test statistic such that if the observed of the test statistic falls in
the critical region we will reject
H
0
and accept
H
1
.
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Type I and II errors
If we reject
H
0
when
H
0
is true, we say that we have committed an error of
type I
and
α
=
P
(type I error) =
P
( reject
H
0
when
H
0
is true)
If the observed value of the test statistic does not fall in the critical region,
then we fail to reject
H
0
. If we fail to reject
H
0
when
H
0
is false, then we
say that we have committed an error of
type II
and
β
(
θ
1
)
=
P
(type II error)
=
P
( fail to reject
H
0
when
θ
=
θ
1
∈
H
1
)
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