Download Chapter 1 Hypothesis Testing

Topics
Chapter 1
Hypothesis Testing
Hypothesis Testing –
Two Samples
Econometrics
Prof. Monica Roman
Independent samples Z test for the difference in
two means
Pooled-variance t test for the difference in two
means
F Test for the Difference in Two Variances
References
Homework
1
Econometrics
Prof. Monica Roman
2
Comparing Two Independent
Samples
Examples
Comparing the means for Two Independent
Samples
Is there a difference in the mean value of residential
real estate sold by male agents and female agents in
Bucharest?
Is there a difference in the mean number of days
absent between young workers (under 21 years of
age) and older workers (more than 60 years of age) in
the fast-food industry?
Is there is a difference in the proportion of AES
graduates and WU Timisoara graduates who pass
Econometrics exam at their first attempt?
Is there an increase in production if a new software is
introduced into the production area?
Econometrics
Prof. Monica Roman
Different Data Sources
Unrelated
Independent
3
Sample selected from one population has no effect on
the sample selected from the other population
Use the Difference between 2 Sample Means
Use Z Test or t Test for Independent Samples
Econometrics
Prof. Monica Roman
4
1
Independent Sample Z Test
(Variances Known)
Setting Up the Hypotheses
D- hypothesized difference
H 0: µ 1 = µ 2
H 1: µ 1 ≠ µ 2
H 0: µ 1 - µ 2 = D
H 1: µ 1 - µ 2 ≠ D
OR
H 0: µ 1 ≤ µ 2
H 1: µ 1 > µ 2
H 0: µ 1 - µ 2 ≤ D
H 1: µ 1 - µ 2 > D
OR
H 0: µ 1 ≥ µ 2
H 1: µ 1 < µ 2
H 0:
H 1:
OR
Econometrics
Prof. Monica Roman
µ1 -µ2 ≥D
µ1 -µ2 <D
Two
Tail
Right
Tail
σ1
n1
Samples are randomly and independently drawn
from normal distributions
n1>30 and n2>30
Population variances are known
The mean of sample distribution of (x1-x2)
The standard deviation of Sample distribution of
(x1-x2)
5
Econometrics
Prof. Monica Roman
6
t Test for Independent
Samples (Variances Unknown)
( X 1 − X 2 ) − ( µ1 − µ 2 )
2
Left
Tail
Test Statistic
Z=
Assumptions
+
σ2
2
n2
Rejection rule
Assumptions
Both populations are normally distributed
Samples are randomly and independently drawn
Population variances are unknown but assumed equal
Notice!
If both populations are not normal, need large sample sizes!
Econometrics
Prof. Monica Roman
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Econometrics
Prof. Monica Roman
8
2
Developing the t Test for
Independent Samples
(continued)
Calculate the Pooled Sample Variance as an
Estimate of the Common Population Variance
S : Pooled sample variance
n1 : Size of sample 1
S12 : Variance of sample 1
n2 : Size of sample 2
2
2
S : Variance of sample 2
Econometrics
Prof. Monica Roman
Econometrics
Prof. Monica Roman
(continued)
Compute the Sample Statistic
(X
df = n1 + n2 − 2
S p2 =
9
Standard deviations are
unequal
t=
( n − 1) S12 + (n2 − 1) S 22
S = 1
(n1 − 1) + (n2 − 1)
2
p
2
p
Developing the t Test for
Independent Samples
1
− X 2 ) − ( µ1 − µ 2 )
1 1
S p2  + 
 n1 n2 
Hypothesized
difference
( n1 − 1) S12 + ( n2 − 1) S22
( n1 − 1) + ( n2 − 1)
Econometrics
Prof. Monica Roman
10
Example
The sample standard
deviations s1 and s2 are
used in place of the
respective population
standard deviations.
The degrees of freedom are
adjusted downward by a
rather complex
approximation formula. The
effect is to reduce the
number of degrees of
freedom in the test, which
will require a larger value of
the test statistic to reject the
null hypothesis.
11
You’re a financial analyst for Charles Schwab. Is there
a difference in average dividend yield between stocks
listed on the NYSE & NASDAQ? You collect the
following data:
NYSE
NASDAQ
Sample size
21
25
Sample Mean
3.27
2.53
Sample Std Dev
1.30
1.16
Assuming equal variances, is there a difference in
average yield (α = 0.05)?
Econometrics
Prof. Monica Roman
12
3
Solution
Calculating the Test Statistic
H0: µ1 - µ2 = 0 i.e. (µ
µ1 = µ2 )
H1: µ1 - µ2 ≠ 0 i.e. (µ
µ1 ≠ µ2)
α = 0.05
df = 21 + 25 - 2 = 44
Critical Value(s):
Reject H0
Reject H0
.025
.025
-2.0154 0 2.0154
2.03
t
Test Statistic:
t=
3.27 − 2.53
1 
 1
1.502  + 
 21 25 
= 2.03
Conclusion:
There is evidence of a
difference in means.
1 1
S p2  + 
 n1 n2 
=
( 3.27 − 2.53) − 0
 1 1 
1.510  + 
 21 25 
Econometrics
Prof. Monica Roman
S12
F= 2
S2
Assumptions
14
S 12 = Variance of Sample 1
n1 - 1 = degrees of freedom
S
2
2 = Variance of Sample 2
Both populations are normally distributed.
= 2.03
The F Test Statistic
Parametric Test Procedure
− X 2 ) − ( µ1 − µ 2 )
13
Test for the Difference in 2 Independent
Populations
1
( n1 − 1) S12 + ( n2 − 1) S22
( n1 − 1) + ( n2 − 1)
( 21 − 1)1.302 + ( 25 − 1)1.162 = 1.502
=
( 21 − 1) + ( 25 − 1)
F Test for Difference in Two
Population Variances
(X
S p2 =
Decision:
Reject at α = 0.05.
Econometrics
Prof. Monica Roman
t=
n2 - 1 = degrees of freedom
Test is not robust to this violation!
Samples are randomly and independently drawn.
Econometrics
Prof. Monica Roman
15
0
Econometrics
Prof. Monica Roman
F
16
4
Developing the F Test
Hypotheses
H 0: σ 12 = σ 22
H 1: σ 12 ≠ σ 22
α/2
Do Not
Reject
FL
Easier Way
Reject H0
F = S12 /S22
0
Two Sets of Degrees of Freedom
Reject H0
Test Statistic
Developing the F Test
α/2
Do Not
Reject
Test Statistic
FU
Reject H0
Put the largest in the num.
F = S12 /S22
0
F
α
F
F
df1 = n1 - 1; df2 = n2 - 1
Critical Values: FL(
FL =
1/FU*
)
and FU(
n1 -1, n2 -1
)
n1 -1 , n2 -1
(*degrees of freedom switched)
Econometrics
Prof. Monica Roman
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F Test: An Example
Econometrics
Prof. Monica Roman
18
F Test: Example Solution
Assume you are a financial analyst for Charles Schwab. You want
to compare dividend yields between stocks listed on the NYSE &
NASDAQ. You collect the following data:
NYSE
NASDAQ
Number
21
25
Mean
3.27
2.53
Std Dev
1.30
1.16
Finding the Critical
Values for α = .05
df1 = n1 − 1 = 21 − 1 = 20
df 2 = n2 − 1 = 25 − 1 = 24
F.05,20,24 = 2.03
Is there a difference in the
variances between the NYSE
& NASDAQ at the α = 0.05 level?
Econometrics
Prof. Monica Roman
19
© 1984-1994 T/Maker Co.
Econometrics
Prof. Monica Roman
20
5
F Test: Example Solution
F Test: One-Tail
Test Statistic:
H0: σ12 = σ22
2
Critical Value(s):
.05
2.33
1.25
H1: σ12 < σ22
Decision:
Do not reject at α = 0.05.
Reject
0
H0: σ12 ≥ σ22
S 2 1.302
F = 12 =
= 1.25
S2 1.162
H1: σ1 ≠ σ2
α = .05
df1 = 20 df2 = 24
2
F
Conclusion:
There is insufficient
evidence to prove a
difference in variances.
Econometrics
Prof. Monica Roman
Reject
1
FU ( n2 −1, n1 −1)
Degrees of
freedom
switched
Reject
0
α = .05
F
F L ( n1 − 1, n 2 − 1 )
0
Econometrics
Prof. Monica Roman
FU ( n1 −1, n 2 − 1)
F
22
Homework & References
Voineagu, V. si colectiv- Teorie si practica
econometrica, Ed. Meteor Press, 2007,
pages 100-110
Read the text and solve the exercises!
Reject H0
Put the largest in the num.
Do Not
Reject
Test Statistic
FL ( n1 −1, n2 −1) =
21
Easier Way
α = .05
H0: σ12 ≤ σ22
H1: σ12 > σ22
α = .05
F Test: One-Tail
or
F = S12 /S22
0
Econometrics
Prof. Monica Roman
α
F
F
23
Econometrics
Prof. Monica Roman
24
6