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381
Hypothesis Testing
(Introduction-I)
QSCI 381 – Lecture 25
(Larson and Farber, Sect 7.1)
Illustrative Example-I
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It is claimed that 20% of a certain
species of rockfish spawns each year. We
sample 780 fish and find 125 show
evidence for spawning. Is the sample
statistic ( pˆ  125 / 780  0.160 ) “sufficiently
different” from the claim that we can reject
the claim?
Illustrative Example-II
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We construct the sampling distribution
for samples of size 780 if the population
proportion was really 0.2.
data
15
10
0
5
Density
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25
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0.10
Is the difference “big enough”?
0.15
0.20
0.25
Claim
0.30
Sampling Distributions
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A
is a (continuous)
probability distribution for a sample
statistic (given values for the population
parameters).
One can think of the sampling
distribution for the sample mean as a
histogram for the value the sample
mean could take given the collection of
(very) many samples of size n.
Hypothesis Tests-I
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A
is a process that
uses sample statistics to test a claim
about the value of a population
parameter.
Colloquially, what we are going to do is
to see whether the data we have “could
have happened” if the claim was true.
Hypothesis Tests-II
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A claim about a population parameter is
called a
. To test
a statistical hypothesis, we state a pair
of hypotheses – one that represents the
claim and the other its complement.
We use statistical methods to determine
whether or not we can reject the claim.
Hypothesis Tests-III
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A
H0 is a statistical
hypothesis that contains a statement of
equality, e.g. , , or =.
The
is the
complement of the null hypothesis. It is a
statement that must be true if H0 is false.
H0 is read as “H subzero” or “H naught”, Ha is
read as “H sub-a”.
Note that it is sometimes necessary to define
the claim as the alternative hypothesis.
Null and Alternative Hypotheses
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First determine the claim and hence the
null hypothesis.
H0 :   k;
Ha :   k
H0 :   k;
Ha :   k
H0 :   k;
Ha :   k
The first two hypotheses have one-sided alternatives
while the third hypothesis has a two-sided alternative.
Null and Alternative Hypotheses
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What are the null and alternative
hypotheses for:
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The density of salmon is 100 fish / ha.
The escapement is 20%.
More than 60% of the population is
mature.
The number of recaptures is 7.
The number of recaptures is not 7.
Null and Alternative Hypotheses
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Notes:
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The null hypothesis and the alternative
hypothesis should be determined before
the data are collected.
Always use a two-sided alternative unless
there are good theoretical reasons for
using a one-sided alternative. This is
particularly true if the hypotheses are
constructed after the data are collected.
Testing Hypotheses
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To test a hypothesis:
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There are two outcomes from this:
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We assume the null hypothesis is true.
We determine the sampling distribution for the
data.
We compare the data with the sampling
distribution for the data if the null hypothesis was
true.
We reject the null hypothesis.
We fail to reject the null hypothesis.
Note: we do not accept the null hypothesis –
why not?
Type I and Type II Errors-I
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A
occurs if the null
hypothesis is rejected when it is actually
true.
A
occurs if the null
hypothesis is not rejected when it is
actually false.
Errors occur because the sample is not
the same as the population.
381
Type I and Type II Errors-II
Actual Truth of H0
Decision
H0 is true
Do not reject H0 Correct decision
Reject H0
Type I error
(Prob=)
H0 is false
Type II error
(Prob = )
Correct decision
(Power = 1-)
Type I and Type II Errors-III
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The consequences of Type I and Type II errors can
be quite different and very important. Consider the
claims:
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By implementing this management measure, the rate of
recovery will be at least 1% per annum.
We sampled 4 clams and claim that the proportion of the
population with a given disease is 3% or less.
We claim that male and female gnu grow at the same rate.
We measure 10,000 gnu and compare the mean lengths of
males and females.
How would you balance Type I and Type II errors in
these cases.
Significance
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In a hypothesis test, the
is the maximum allowable
probability of making a type I error
(denoted ).
The probability of making a type II
error is denoted by .
There are three commonly used levels
of significance (=0.1, 0.05 and 0.01) –
they are all arbitrary.