M155 L15: Hypotheses and Events Slide 1

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M155 L15 Hypotheses and Events -- Slide 1
POPULATION SAMPLE PARAMETER
STATISTIC mean ? variance ?2
s2 standard deviation ? s fraction p sum
Np X
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M155 L15 Hypotheses and Events -- Slide 2
(i) p gt 0.4 p ? 0.4 (ii) p lt 0.4 p
? 0.4 (iii) p ? 0.4 p 0.4
H1 p gt 0.4 p lt 0.4 p ? 0.4
H0 p ? 0.4 p ? 0.4 p 0.4
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CAUTION A statement like p gt 4 is not a
reasonable hypothesis about p, because the
integer 4 is not a possible value of p. And X gt
4 is not a hypothesis at all, since it talks
about the sample sum X.   To convince us that the
alternative hypothesis is true, the sample
evidence must be inconsistent with the null
hypothesis. In most cases, we will use the
phrase "at what level is the evidence" to
announce the alternative hypothesis, which in
Examples A,B,C below will be italicized .
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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, at what level is the
evidence that the fraction (of all voters in
favor of raising taxes) is larger this
year? RIGHT-SIDED H1 p gt 0.4 H0 p ?
0.4 n8 sample event X 6 supporting
event X ? 6 Example A has an alternative
hypothesis pgt0.4 corresponding to the phrase the
fraction is larger. Examples A and B each have a
supporting event with an inequality sign in the
same direction as the inequality sign in the
alternative hypothesis.
M155 L15 Hypotheses and Events -- Slide 3
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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. At what
level is the evidence that the amendment will
receive too little support to pass? LEFT-SIDED
H1 p lt 0.7 H0 p ? 0.7 n8 sample
event X 3 supporting event X ?3 Example B
has an alternative hypothesis plt0.7 corresponding
to the phrase too little support.
M155 L15 Hypotheses and Events -- Slide 4
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EXAMPLE C An bill is hotly debated in the
legislature. If 7 voters in a random sample of
8 favor the bill, at what level is the evidence
that the fraction in favor is different from
50?   TWO-SIDED H1 p ? 0.5 H0 p ?
0.5 n8 sample event X 7
supporting event X ? 1 OR X ? 7 Example
C has an alternative hypothesis p?0.5
corresponding to the phrase the fraction is
different. CAUTION MANY writers will always
write H0 as the simple equality shown in Example
C, even if H1 is one-sided.
M155 L15 Hypotheses and Events -- Slide 5
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Example C is different, because the alternative
hypothesis p ? 0.5 has two sides plt0.5 and
pgt0.5 this means we need a supporting event
with two sides X ? 1 supports plt0.5, while X ?
7 supports pgt0.5 .   But where does the 1
come from? Computations for a two-sided
alternative hypothesis can be very complicated
unless the null hypothesis is p0.5 . Under
this null hypothesis, the distribution of this
binomial random variable is symmetric about the
level Xn/2 (which means X4 in Example C).
Thus it is just as likely that the count X be at
least three units above 4 (i.e., X?7) as it is
likely to be at least three units below 4 (i.e.,
X?1). The only information in the data is that
the sample count X is 3 units from 4.
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End of Lesson 15
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1.2 An Example   Suppose you are concerned about
improving the school system where you live, and
it appears taxes would have to be raised in order
to make the improvements. Last year you got a
referendum on the ballot to raise taxes, but it
was defeated only 40 of the voters in the
district approved it. This year you would like
to try again, but before you can decide what to
do, you would like to have some evidence of an
increase in the fraction of voters favorable to
raising taxes. If there has been some increase,
you are willing to run a campaign to gather
enough support to win. Otherwise, you feel it is
not worthwhile.
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• 1.3 Two-Part Populations
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• In Chapter Three, we computed the probability of
  features of a sample to be taken in the future,
  and the tool we used was the binomial
  distribution for a two-part population. A
  two-part population is a population whose
  subjects are considered in two categories the
  counted category and its complement. Here are
  some examples of counted categories which might
  occur in practice.
•  
• (1) voters who want to raise taxes
• hemopheliacs for whom a certain blood coagulant
  drug
• is effective
• (3) legislators who want to approve a certain
  amendment
• (4) auto fatality cases in Maryland which are
  alcohol-related

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2.1 Hypotheses that Compare   A hypothesis is a
statement about a population, often involving a
parameter of the population (like p or m).
Hypotheses come in complementary pairs, so one
hypothesis in the pair must be true, but we don't
know which one. Here are some examples of
hypothesis pairs.   (i) p gt .4 p ? .4
(ii) p lt .4 p ? .4 (iii) p ? .4 p .4   The
purpose of a hypothesis test is to evaluate the
evidence for one of the hypotheses in a pair, and
decide at what level that evidence is. The
hypothesis we are trying to establish is called
the research hypothesis, but more often called
the alternative hypothesis H1. Its complement
is called the null hypothesis H0. Other symbols
for the alternative hypothesis are HA and Ha .
But do not use Ha it looks like H0.
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As a matter of convenience, the alternative
hypothesis usually expresses a strict inequality,
like one of these   H1 p gt .4 p lt .4 p ?
.4   while the null hypothesis has equality as
part of it, like one of these   H0 p ? .4 p
? .4 p .4   In designing a hypothesis test,
first we decide what the parameter is, and then
we write a complementary pair of hypotheses.
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2.2 Events that Support the Alternative
Hypothesis   Suppose we study a pair of
hypotheses with a random sample of n voters and
record the count x of voters who favor raising
taxes. Generally speaking if x is very large,
then the data strongly support the hypothesis
pgt.4 but if x is very small, then the data
strongly support the hypothesis plt.4 and if x
is near 40 of n, then the data strongly support
the hypothesis p.4 . But words like "strongly
support" are too vague for statistics. To make
these statements quantitative, we imagine
choosing a second random sample of size n . The
data we gathered may be considered an occurrence
of the sample event X x .   To be at least
as convincing as the sample event (that the
alternative hypothesis is true), what event would
have to occur in our imaginary second experiment?
We call that the supporting event, and we will
see that the supporting event depends on the
alternative hypothesis.