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1
Introduction to Probability
and Statistics
Thirteenth Edition
Chapter 9
LargeSample Tests of
Hypotheses
Introduction
• Suppose that a pharmaceutical company
is concerned that the mean potency
m
of an
antibiotic meet the minimum government
potency standards. They need to decide
between two possibilities:
–
The mean potency
m
does not exceed
the mean allowable potency.
–
The mean potency
m
exceeds the
mean allowable potency.
•This is an example of a
test of hypothesis.
Introduction
• Similar to a courtroom trial. In trying a
person for a crime, the jury needs to
decide between one of two possibilities:
–
The person is guilty.
–
The person is innocent.
• To begin with, the person is assumed
innocent.
• The prosecutor presents evidence, trying
to convince the jury to reject the original
assumption of innocence, and conclude
that the person is guilty.
Parts of a Statistical Test
1. The null hypothesis, H
0
:
–
Assumed to be true until we can
prove otherwise.
2. The alternative hypothesis, H
a
:
–
Will be accepted as true if we can
disprove H
0
Court trial:
Pharmaceuticals:
H
0
: innocent
H
0
:
m
does not exceeds allowed
amount
H
a
: guilty
H
a
:
m
exceeds allowed amount
Parts of a Statistical Test
3.
The test statistic and its
p
value:
•
A single statistic calculated from the
sample which will allow us to reject or not
reject H
0
, and
•
A probability, calculated from the test
statistic that measures whether the test
statistic is
likely
or
unlikely
, assuming H
0
is true.
4.
The rejection region:
–
A rule that tells us for which values of
the test statistic, or for which
p
values,
the null hypothesis should be rejected.
Parts of a Statistical Test
5.
Conclusion:
–
Either ―Reject H
0
‖ or ―Do not reject
H
0
‖, along with a statement about the
reliability of your conclusion.
How do you decide when to reject H
0
?
–
Depends on the
significance level,
a,
the maximum tolerable risk you
want to have of making a mistake, if
you decide to reject H
0
.
–
Usually, the significance level is
a =
.01
or
a = .05.
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Example
•
The mayor of a small city claims that the
average income in his city is $35,000 with
a standard deviation of $5000. We take a
sample of 64 families, and find that their
average income is $30,000. Is his claim
correct?
12.
We want to test the hypothesis
:
H
0
:
m
= 35,000 (mayor is correct) versus
H
a
:
m
35,000 (mayor is wrong)
Start by assuming that H
0
is true and
m
= 35,000.
Example
3. The best estimate of the population mean
m
is the
sample mean, $30,000
:
•
From the Central Limit Theorem the sample mean
has an approximate normal distribution with mean
m
= 35,000 and standard error SE = 5000/8 = 625.
•
The sample mean, $30,000 lies
z
= (30,000 –
35,000)/625 = 8 standard deviations below the
mean.
•
The probability of observing a sample mean this far
from
m
= 35,000 (assuming H
0
is true) is
nearly
zero
.
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 Fall '08
 Any
 Statistics, Probability

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