Download Chapter 1 Hypothesis Testing

Chapter 1
Hypothesis Testing
Principles
of Hypothesis Testing
tests for one sample case
Econometrics
Prof. Monica Roman
1
Statistical Hypotheses
They are defined as assertion or conjecture about the parameter
or parameters of a population, for example the mean or the
variance of a normal population.
They may also concern the type, nature or probability distribution
of the population.
Statistical hypotheses are based on the concept of proof by
contradiction.
For example, say, we test the mean (m) of a population to see if
an experiment has caused an increase or decrease in m. We do
this by proof of contradiction by formulating a null hypothesis.
Econometrics
Prof. Monica Roman
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Elements of statistical
hypothesis
There are five ingredients to any statistical
test :
(a) Null Hypothesis
(b) Alternate Hypothesis
(c) Test Statistic
(d) Rejection/Critical Region
(e) Conclusion
Econometrics
Prof. Monica Roman
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Null Hypothesis
It is a hypothesis which states that there is no
difference between the procedures and is denoted
by H0.
For the above example the corresponding H0 would
be that there has been no increase or decrease in
the mean.
Always the null hypothesis is tested, i.e., we want to
either accept or reject the null hypothesis because
we have information only for the null hypothesis.
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Prof. Monica Roman
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Alternative Hypothesis
It is a hypothesis which states that there is a
difference between the procedures and is
denoted by H1.
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Prof. Monica Roman
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Various types of H0 and H1
Null Hypothesis
Case
H0
Alternate Hypothesis
H1
1
µ1 = µ2
µ1 ≠ µ2
2
µ1 < µ2
µ1 > µ2
3
µ1 > µ2
µ1 < µ2
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Prof. Monica Roman
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Reason for Rejecting H0
Sampling Distribution
It is unlikely
that we would
get a sample
mean of this
value ...
... Therefore, we
reject the null
hypothesis that
µ = 50.
... if in fact this were
the population mean.
20
µ = 50
H0
Econometrics
Prof. Monica Roman
Sample Mean
7
Test Statistic
It is the random variable X whose value is tested to arrive at a
decision.
The Central Limit Theorem states that for large sample sizes (n >
30) drawn randomly from a population, the distribution of the
means of those samples will approximate normality, even when the
data in the parent population are not distributed normally.
A z statistic is usually used for large sample sizes (n > 30)
but often large samples are not easy to obtain, in which case the tdistribution can be used. The population standard deviation s is
estimated by the sample standard deviation, s. The t curves are
bell shaped and distributed around t=0.
t
=
X
S
−
µ
n
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Prof. Monica Roman
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Rejection Region
It is the part of the sample space (critical region) where the null
hypothesis H0 is rejected.
The size of this region, is determined by the probability (α) of the
sample point falling in the critical region when H0 is true. α is also
known as the level of significance, the probability of the value
of the random variable falling in the critical region.
Also it should be noted that the term "Statistical significance"
refers only to the rejection of a null hypothesis at some level α. It
implies only that the observed difference between the sample
statistic and the mean of the sampling distribution did not occur
by chance alone.
Econometrics
Prof. Monica Roman
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Conclusion
Conclusion : If the test statistic falls in the
rejection/critical region, H0 is rejected, else
H0 is accepted.
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Prof. Monica Roman
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Errors
Type I Error
Reject True Null Hypothesis (“False Positive”)
Has Serious Consequences
Probability of Type I Error Is α
Called Level of Significance
Set by researcher
Type II Error
Do Not Reject False Null Hypothesis (“False Negative”)
Probability of Type II Error Is β (Beta)
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Prof. Monica Roman
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Errors
Test Result –
True State
H0 True
H0 False
H0 True
H0 False
Correct
Decision
Type I Error
Type II Error
Correct
Decision
α = P(Type I Error ) β = P (Type II Error )
Goal: Keep α, β reasonably small
Econometrics
Prof. Monica Roman
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Level of Significance, α and
the Rejection Region
α
H0: µ ≥ 3
H1: µ < 3
H0: µ ≤ 3
H1: µ > 3
Rejection
Regions
0
0
H0: µ = 3
H1: µ ≠ 3
Critical
Value
α
α/2
0
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Prof. Monica Roman
13
Steps
1.
2.
State the null and alternative hypotheses
Choose α. The value should be small,
usually less than 10%. It is important to
consider the consequences of both types of
errors.
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Prof. Monica Roman
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3.
Select the test statistic.
Determine its value from the sample data.
This value is called the observed value of the
test statistic.
! Remember that a t statistic is usually
appropriate for a small number of
samples; for larger number of samples, a
z statistic can work well if data are
normally distributed.
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Prof. Monica Roman
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4
Compare the observed value of the statistic
to the critical value obtained for the chosen α.
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Prof. Monica Roman
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5.
Make a decision:
If statistic falls in the rejection region
Reject H0 in favour of H1.
If the test statistic does not fall in the
critical region:
Conclude that there is not enough evidence
to reject H0.
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Prof. Monica Roman
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Hypothesis Testing on mean
t-Test: s Unknown
Assumptions
Population is normally distributed
If not normal, only slightly skewed & a large
sample taken (Central limit theorem applies)
Parametric test procedure
t test statistic, with n-1 degrees of freedom
X −µ
t =
S
n
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Prof. Monica Roman
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Hypothesis Testing: Steps
Test the Assumption that the true mean of
monthly cinema attendance of students is
at least 3.
1. State H0
H0 : µ ≥ 3.0
2. State H1
H1 : µ < 3.0
3. Choose α
α = .05
4. Choose n
n = 25
5. Choose Test:
t Test (or p Value)
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Prof. Monica Roman
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Hypothesis Testing: Steps
(continued)
6. Set Up Critical Value(s)
7. Collect Data
8. Compute Test Statistic
t = -1.7
25 students sampled, mean=2.7, s=0.75
Computed Test Stat.= -2
(computed P value=.04, two-tailed test)
9. Make Statistical Decision
10. Express Decision
Reject Null Hypothesis
The true mean is less than 3.0
Econometrics
Prof. Monica Roman
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Hypothesis Testing on σ2
Two-tailed test
Ho: σ2 = σ2o
Ha: σ2 ≠ σo2
(n - 1)s2
Test statistic: χ2 =
σo2
One tail test
2
Ho: σ2 >= σ
o
Ho: σ2 <= σ2o
Ha: σ2 > σo2
Ha: σ2 < σo2
reject Ho if χ2* > χα2 , n-1
reject Ho if χ2* < χ2α, n-1
Econometrics
Prof. Monica Roman
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Homework & References
1. Voineagu, V. si colectiv- Teorie si practica econometrica,
Ed. Meteor Press, 2007, pages 85-98
Read the text and solve the exercises.
2. David Ray Anderson,Dennis J. Sweeney,Thomas Arthur
Williams,Thomas A. Williams - Statistics for business and
economics, Chapter 9.
http://books.google.ro/books?id=_sOSAXTuy8cC&pg=PA388&l
pg=PA388&dq=hypothesis+testing+textbook+sweeney&sou
rce=bl&ots=24jbOAksnM&sig=xfPsCcF1GCwaLIZYA7bDJsP
z80k&hl=ro&ei=1s29TK2oMoOhOsSGyGY&sa=X&oi=book_r
esult&ct=result&resnum=1&ved=0CBQQ6AEwAA#v=onepag
e&q&f=false
Econometrics
Prof. Monica Roman
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