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493
Chapter 10
Testing Claims Regarding a Parameter
10.1 The Language of Hypothesis Testing
1.
A Type I error is the error of rejecting
0
H
when in fact
0
H
is true.
A Type II error is the
error of
not
rejecting
0
H
when in fact
1
H
is true.
2.
The probability of making a Type I error is
α
so we should choose
01
.
0
=
to minimize
the chance that we make a Type I error.
3.
As we decrease
(the probability of rejecting a true
0
H
), we are effectively making it less
likely that we will reject
0
H
since we require stronger evidence against
0
H
as
decreases.
This means that it is also more likely that we will fail to reject
0
H
when
1
H
is
really true, so
β
increases. Thus,
increases as
decreases.
4.
Probability of Type I error
0.05
=
=
.
5.
In a hypothesis test we make a judgement about the validity of a hypothesis based on the
available data.
If the data contradicts
0
H
then we reject
0
H
.
However, if the available
data do not contradict
0
H
, this does not guarantee that
0
H
is true. Consider the court
system in the U.S., where suspects are assumed to be innocent until proven guilty. An
acquittal does not mean the suspect is innocent, merely that there was not enough evidence
to reject the assumption of innocence.
6.
Reasonable doubt is based on logic and reasoning. The phrase “beyond all reasonable
doubt” means that there is sufficient and convincing evidence that a reasonable person
would not hesitate to act upon it. It is not based on circumstantial evidence or
unsubstantiated accusations. The primary difference is that “beyond all doubt” expects
every possibility to be considered while “beyond all reasonable doubt” expects that all
reasonably likely possibilities have been taken into account.
7.
False; sample evidence will never prove a null hypothesis is true. We assume the null is
true and try to gather enough evidence to say that the null is not true. Failing to reject a null
hypothesis does not imply that the hypothesis is actually true, just that there was not
enough evidence to reject the assumption that it is true.
8.
False; decreasing the probability of making one type of error will increase the probability
of making the other type (see #3 above).
9.
Parameter
μ
=
.
Righttailed since
1
:5
H
>
10.
Parameter
p
=
.
Lefttailed since
1
:0
.
2
Hp
<
11.
Parameter
σ
=
.
Twotailed since
1
:4
.
2
H
≠
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Testing Claims Regarding a Parameter
494
12.
Parameter
p
=
.
Righttailed since
1
:0
.
7
6
Hp
>
13.
Parameter
μ
=
.
Lefttailed since
1
:
120
H
<
14.
Parameter
σ
=
.
Twotailed since
1
:7
.
8
H
≠
15.
(a)
0
:
0.118
=
,
1
:
0.118
<
The alternative hypothesis is < because the sociologist believes the percent has
decreased.
(b)
We make a Type I error if the sample evidence leads us to reject
0
H
and believe that
the percentage of registered births to teenage mothers is less than 11.8% when, in fact,
it is not less than 11.8%.
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 Spring '11
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