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3. REJECTING THE NULL HYPOTHESIS H0   Ideally,
we would like to reject the null hypothesis H0
when it is false and accept it when it is true.
But we usually do not know whether the null
hypothesis is actually true, and we must make a
reasonable decision on probabilistic grounds.
The sample is different from the population
from which it is drawn. This difference is
called sampling error, and it can mislead us to
make a wrong decision about the null hypothesis
H0 .
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3.1 The p-value   The observed level of
significance or p-value is the probability of the
supporting event at the edge of the null
hypothesis. That means that if the null
hypothesis were actually true with the edge value
p0 of the parameter p, and if the same size
random sample were collected again, then the
p-value is the probability that the supporting
event would occur again. CAUTION the p-value
is not a value of the population fraction p nor
of any other parameter.   The p-value measures
how consistent the sample is with the null
hypothesis. If the p-value is large, then the
supporting event was likely to have occurred
under the null hypothesis, and we will not reject
H0 . But if the p-value is small, then the
supporting event was not likely to have occurred
under the null hypothesis, and we will reject H0
.
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3.2 Decision Rules How Small is
"Small"?   Before the sample is chosen, the
decision-maker defines a "small" observed level
(p-value), by specifying a maximum level of
significance a (pronounced alpha), implying
this decision rule   If p-value ? a, reject H0
in favor of H1 (act as though H1 were true). If
p-value gt a, do not reject H0 .   If the null
hypothesis is rejected, then H1 has been
established but if the null hypothesis is not
rejected, beware that H0 has not been
established.
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In principle, the specified level a can be any
number between 0 and .5, but in practice a is
one of the numbers .01, .05, .10, .20 and the
most common value is .05 . Therefore we can
expect to reject the null hypothesis when the
p-value is .005 and accept the null hypothesis
when the p-value is .25, even if no one has told
us what a is. "There is evidence at all
levels" means the observed level was zero to the
number of decimal places that it was calculated.
"The data are not significant" means we did not
reject the null hypothesis the observed level
was bigger than the specified level a (even if we
are not told what a is).
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Compare McClave X6.29
For each p-value and ?, would you reject the null
hypothesis? a. ? .10 and pvalue .06 b. ?
.05 and pvalue .06 b'. ? .05 and pvalue
.05 c. ? .01 and pvalue .02 d. ? .025
and pvalue .02 e. ? unknown and pvalue
.23 f. ? unknown and pvalue .005
a,d,f
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We reject the null hypothesis at the p-value and
at every higher level of significance. If
p-value ? a, we would say that that we reject
the null hypothesis at the a level. If we
reject the null hypothesis at the .05 level, we
may not know what the p-value is all we know is
that p-value ? .05 .   EXAMPLE D (Cf.
Example B.) An amendment to a certain state's
constitution must be approved by 70 of
legislators to become law. A random sample of 8
legislators showed only 3 in favor. Is there
evidence at the 10 level that the amendment will
receive too little support to pass? Is there
evidence at the 5 level? Is there evidence at
the 1 level?
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SOLUTION D State the basic conclusion with the
observed level of significance (p-value). Then
we append a short sentence for each specified
level of significance. At the .058 level,
there is evidence that less than 70 of the
legislators favor the amendment. There is also
evidence at the 10 level. There is NOT evidence
at the 5 level. There is NOT evidence at the 1
level.   WARNING 1 ALWAYS give the basic
conclusion first, (even if it is not "asked")
using the observed level or p-value. WARNING 2
Do NOT put two levels of significance in the
same sentence. It can lead to statements like
"At the .058 level, there is evidence at the 10
level ", which is MEANINGLESS!
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3.3 Decision Errors   If we reject the null
hypothesis when it is true, we make a mistake
called a Type I Error. As we have seen above,
the probability of doing so is the specified
level a. Therefore a is called the probability
of a Type I Error. You cant make a Type I
Error unless the null hypothesis is actually
true. In general, the choice of a depends on
the costs of making different kinds of error.
But the decision maker often chooses a on the
basis of practical experience, striking a balance
among different concerns. The smaller we make a,
the less likely we are to reject the null
hypotheses, even when it is false.
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If we fail to reject the null hypothesis when
it is false, we make a mistake called a Type II
Error. The probability of doing so is called ?
(pronounced beta). We will not compute the
probability ? of a Type II Error, because it
depends on the unknown true value of the
population parameter. We need only observe that
making a smaller unfortunately makes ? larger.
You can't make a Type II Error unless the null
hypothesis is actually false.
X
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EXERCISE 13 Define the term Type I Error. In a
separate sentence, explain what must be true
before a Type I Error is even possible. Give the
mathematical symbol for the probability of a Type
I Error.   EXERCISE 14 Define the term Type II
Error. In a separate sentence, explain what must
be true before a Type II Error is even possible.
Give the mathematical symbol for the probability
of a Type II Error.
X
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3.4 Power of a Test   It is possible that the
alternative hypothesis be true, and yet the data
look consistent with the null hypothesis. This
usually happens because the actual value of the
population parameter is close to the edge of the
null hypothesis. In fact, an alternative
hypothesis usually contains many possibilities
for many possible values of the parameter under
study. Thus in Example A, the alternative
hypothesis pgt.4 contains the possibilities
p.41, p.42, p.5, p.6, and many others. Some
of these would easily give rise to sample data
very consistent with p.4, which is part of the
null hypothesis p?.4 .
H1 p gt .4
H0 p lt .4
p
.4
X
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The ability to detect such a circumstance and
establish an alternative hypothesis close to the
null hypothesis is called the power of a test.
Power is defined precisely as the probability of
rejecting the null hypothesis when it is
false. Power depends on the size of the sample,
the scale of the variable, and any special
features of the population distribution which may
be invoked by the test. A large sample can
detect very small deviations from the null
hypothesis. Such small deviations are
considered statistically significant, but need
not be practically significant. So it is
important to have a sense for the size of effect
that the test can discern. When inferring the
value of a population parameter, use the most
powerful test which applies to the population and
sample.
X
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3.5 Testing the Null Hypothesis H0   This
subsection talks about a different way to word
the question and solution, and we write the null
hypothesis first. But the hypothesis we use and
the data we use and the observed level (p-value)
we calculate are the same as before.   Problems
in this book will emphasize the alternative
hypothesis by asking, "at what level is the
evidence that the alternative hypothesis is
true". Consistent with this, our conclusions
state that "at the level of significance,
there is evidence that the alternative
hypothesis is true".
X
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EXAMPLE A' Suppose 40 of the voters wanted to
raise property taxes last year, to improve the
school system. If 6 voters in a random sample of
8 want to raise taxes, test the hypothesis that
the fraction (of all voters in favor of raising
taxes) is no larger this year. RIGHT-SIDED H1
p gt .4 H0 p ? .4 At the .050 level of
significance, we reject the hypothesis that no
more than 40 of the voters want to raise taxes
this year.   EXAMPLE B ' An amendment to a
certain state's constitution must be approved by
70 of legislators to become law. A random
sample of 8 legislators showed only 3 in favor.
Test the hypothesis that the amendment will
receive enough support to pass. LEFT-SIDED H1
p lt .7 H0 p ? .7 At the .058 level of
significance, we reject the hypothesis that at
least 70 of the legislators favor the
amendment.  
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EXAMPLE B ' An amendment to a certain state's
constitution must be approved by 70 of
legislators to become law. A random sample of 8
legislators showed only 3 in favor. Test the
hypothesis that the amendment will receive enough
support to pass. LEFT-SIDED H1 p lt .7 H0
p ? .7 At the .058 level of significance, we
reject the hypothesis that at least 70 of the
legislators favor the amendment.   EXAMPLE C '
An bill is hotly debated in the legislature. If
6 voters in a random sample of 8 favor the bill,
test the hypothesis that the fraction in favor is
exactly 50 . omitting exactly may suggest we
mean at least 50   TWO-SIDED H1 p ?
.5 H0 p ? .5 At the .070 level of
significance, we reject the hypothesis that the
fraction of legislators in favor is exactly 50.