Lecture Notes
Chapter Eight: Inferences Based on a Single Sample: Test of Hypothesis
Randall Miller
1 
P a g e
1.
The Elements of a Test of Hypothesis
Conclusions and Consequences for a Test of Hypothesis
True State of Nature
Conclusion
0
H
True
a
H
True
Accept
0
H
(Assume
0
H
True)
Correct decision
Type II error (probability
β
)
Reject
0
H
(Assume
a
H
True)
Type I error (probability
α
)
Correct decision
Elements of a Test of Hypothesis
1.
Null hypothesis
(
0
H
):
A theory about the values of one or more population parameters.
The
theory generally represents the status quo, which we adopt until it is proven false.
By
convention, the theory is stated as
0
H
: parameter = value.
2.
Alternative hypothesis
(
a
H
):
A theory that contradicts the null hypothesis.
The theory
generally represents that which we will accept only when sufficient evidence exists to
establish its truth.
3.
Test statistic
:
A sample statistic used to decide whether to reject the null hypothesis.
4.
Rejection region
:
The numerical values of the test statistic for which the null hypothesis will
be rejected.
The rejection region is chosen so that the probability is
α
that it will contain the
test statistic when the null hypothesis is true, thereby leading to a Type I error.
The value of
α
is usually chosen to be small (e.g., .01, .05, or .10) and is referred to as the
level of
significance
of the test.
5.
Assumptions
:
Clear statements of any assumptions made about the population(s) being
sampled.
6.
Experiment and calculation of test statistic
:
Performance of the sampling experiment and
determination of the numerical value of the test statistic.
7.
Conclusion
:
a.
If the numerical value of the test statistic falls into the rejection region, we reject the
null hypothesis and conclude that the alternative hypothesis is true.
We know that
the hypothesistesting process will lead to this conclusion incorrectly (a Type I error)
only
100
%
α
of the time when
0
H
is true.
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 Fall '08
 STAFF
 Statistics, Null hypothesis, Statistical hypothesis testing

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