Download Hypothesis testing

Hypothesis testing
Another judgment
method of sampling data
Hypothesis
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Null hypothesis
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Simple hypothesis:
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If a population having a distribution function Fθ, where θ is
unknown, and suppose we want to test a specific hypothesis about
θ such as θ=1. We denote this hypothesis H0 and call it the null
hypothesis
The word “null” means we can not assert confidentially this
hypothesis approaches to be true until it is tested by the observed
data set.
Completely specify the population distribution when null hypothesis
is true. E.g., H0: θ=1
Composite hypothesis
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Don’t specify the population distribution when null hypothesis is
true. E.g., H0: θ≧1
Critical region
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The criteria for rejecting or accepting the
null hypothesis
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Specify the specific range or the upper/lower
bound of a certain statistics. Denote the
critical region, C.
If the resultant sample make the calculated
statistics lied in C, then we will reject H0,
otherwise, accept H0.
Type I error and type II error
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Type I
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The test incorrectly calls for rejecting H0
when it is indeed correct.
Prob. (reject H0/H0 is true)
Type II
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The test calls for accepting H0 when it is
false.
Prob. (accept H0/H0 is false)
Significance level
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H0 should only be rejected if the resultant data
are very unlikely when H0 is true.
The significance level α, commonly chosen value
being .01, .05, is the error probability that H0 is
rejected when H0 is indeed true.
The α, in other words, is the maximum type I
error that the tester could endure.
If the error probability of rejecting H0 is greater α,
then the researcher had better to accept H0.
“Far away” from the value of
H0
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H0: θ≦ω (θ is the unknown parameter and
ω is a particular value)
If the estimator of θ, d(X), is far away
from the specified region ω, then we need
to consider the rejection of H0 and
moreover to justify whether the
appropriate significance level α is.
Alternative hypothesis
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We are supposed to be interested in
testing the null hypothesis, H0:θ=θ0,
against the alternative hypothesis,
H1:θ≠θ0, where θ0 is some specified
constant.
H0 (stated the current situation) in contrast to
H1 (stated the experimental/desired situation)
Testing the mean of a normal population—case of
known variance, H0: μ=μ0 vs. H1:μ≠μ0
Critical region
∵
∴
∴
The test statistics distribution
when H0 is true
P-value of a hypothesis test
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If the observed value of test statistics, ν,
lies within the critical region, then the
occurring probability of such a worse ν, we
call it the p-value, must be less than or
equal to α (the significance level).
In other words, if the p-value is smaller
than the chosen significance α, then we will
reject H0.
Operating characteristic (OC)
curve
The OC curve:
the probability of
accepting “H0: μ
=μ0“ under the true
mean is equal to μ
Operating characteristic (OC)
curve (cont.)
• If μ=μ0, then d=0 &
β(μ)=α=0.95.
• If μis very large thanμ0, then
β(μ) will reduce to zero.
The power of test
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The function 1-β(μ) is called the power-function of the test.
The power of the test is equal to the probability of rejection
when μ is the true value.
The more OC β(μ) is, the larger probability of accepting H0
is.
The more power of testing is, the larger probability of
rejection H0 is.
The type II error=prob{accept H0/H0 is false}=β(μ=μ1),
whereμ1≠μ0
Determine the necessary sample size n under the endurable
type II error β, ref. pp.298-299
One-side tests,H0:μ=(≦)μ0 vs.H1:μ＞μ0
The t-test–the case of unknown
variance, H0: μ=μ0 vs. H1:μ≠μ0
Reject when
it is large
The t-test under the unknown
variance
Testing the equality of means of two normal
populations—case of known variances,
H0:μx=μy vs. H1:μx≠μy
In the same way,
H0:μx≦μy vs. H1:μx＞
μy
Testing the equality of means of two normal
populations—case of unknown but equal
variances, H0:μx=μy vs. H1:μx≠μy
Set
∴
In the same way,
Testing the equality of means of two normal
populations—case of unknown and unequal
variances, H0:μx=μy vs. H1:μx≠μy
Even under the case of unknown
and unequal variances
The paired t-test
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Set the new random variable Wi, and
rearrange Wi=Xi-Yi, i=1,2,…n
H0: μw=0 vs. H1: μw≠0
Compute the average of W and use the ttest for hypothesis testing
Hypothesis testing of the variance
of a normal population
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Hypothesis testing for the equality of
variances of two normal populations
Hypothesis tests in Bernoulli
populations
Approximate p testing
Homework #1
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Problems 10,26,28,31,43,55