Matakuliah
Tahun
Versi
: A0064 / Statistik Ekonomi
: 2005
: 1/1
Pertemuan 18
Pembandingan Dua Populasi-2
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Membandingkan pengujian sampel besar
untuk perbedaan antara dua proporsi
populasi dan pengujian untuk kesamaan
dua populasi
2
Outline Materi
• Pengujian Sampel Besar untuk Perbedaan
antara Dua Proporsi Populasi
• Sebaran-F dan Uji untuk Kesamaan Dua
Ragam Populasi
3
COMPLETE
BUSINESS STATISTICS
8-4
5th edi tion
8-5 A Large-Sample Test for the Difference
between Two Population Proportions
•
Hypothesized difference is zero
 I: Difference between two population proportions is 0
•
p1= p2
» H0: p1 -p2 = 0
» H1: p1 -p20
 II: Difference between two population proportions is less than 0
•
•
p1 p2
» H0: p1 -p2  0
» H1: p1 -p2 > 0
Hypothesized difference is other than zero:
 III: Difference between two population proportions is less than D
•
McGraw-Hill/Irwin
p1 p2+D
» H0:p-p2  D
» H1: p1 -p2 > D
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-5
BUSINESS STATISTICS
5th edi tion
Comparisons of Two Population Proportions When
the Hypothesized Difference Is Zero: Test Statistic
When the population proportions are hypothesized to be equal, then a pooled estimator of
the proportion ( ) pmay be used in calculating the test statistic.
A large-sample test statistic for the difference between two population
proportions, when the hypothesized difference is zero:
z
( p1  p 2 )  0
1 1
p(1  p)  
 n1 n2 
x1
x1
is the sample proportion in sample 1 and p
 1  is the sample
n1
n1
proportion in sample 2. The symbol p stands for the combined sample
where p1 
proportion in both samples, considered as a single sample. That is:
pˆ 
McGraw-Hill/Irwin
x x
n n
1
1
1
2
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-6
BUSINESS STATISTICS
5th edi tion
Comparisons of Two Population Proportions When
the Hypothesized Difference Is Zero: Example 8-8
Carry out a two-tailed test of the equality of banks’ share of the car loan market in 1980 and 1995.
Population 1: 1980
n1 = 100
x1 = 53
H 0 : p1  p 2  0
H1: p1  p 2  0
z 
p 1 = 0.53
Population 2: 1995
n 2 = 100
x 2 = 43
p 2 = 0.43
x1 + x 2
53  43
p 

 0.48
n1  n 2 100  100
McGraw-Hill/Irwin
( p1  p 2 )  0
p (1 

 1 1
p ) 


 n1 n2 
0.10

0.004992
0.10

0.53  0.43
 1  1 
 100 100
(.48)(.52) 
 1.415
0.07065
Critical point: z
= 1.645
0.05
H 0 may not be rejected even at a 10%
level of significance.
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-7
BUSINESS STATISTICS
5th edi tion
Example 8-8: Carrying Out the Test
Standard Normal Distribution
0.4
f(z)
0.3
0.2
0.1
0.0
-z0.05=-1.645
Rejection
Region
0
z
z0.05=1.645
Nonrejection
Region
Test Statistic=1.415
McGraw-Hill/Irwin
Rejection
Region
Since the value of the test
statistic is within the
nonrejection region, even at a
10% level of significance, we
may conclude that there is no
statistically significant
difference between banks’
shares of car loans in 1980
and 1995.
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
8-8
5th edi tion
Example 8-8: Using the Template
P-value =
0.157, so do
not reject H0
at the 5%
significance
level.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-9
BUSINESS STATISTICS
5th edi tion
Comparisons of Two Population Proportions When the
Hypothesized Difference Is Not Zero: Example 8-9
Carry out a one-tailed test to determine whether the population proportion of traveler’s check buyers who buy at
least $2500 in checks when sweepstakes prizes are offered as at least 10% higher than the proportion of such
buyers when no sweepstakes are on.
Population 1: With Sweepstakes
n1 = 300
H 0 : p1  p 2  0.10
H 1 : p1  p 2  0.10
x1 = 120
z
p 1 = 0.40
Population 2: No Sweepstakes
n 2 = 700
x 2 = 140
p 2 = 0.20
McGraw-Hill/Irwin

( p 1  p 2 )  D
 p (1  p )
1
 1
 n1 

p (1  p ) 
2
2 
n2
( 0.40  0.20)  0.10
 ( 0.40)( 0.60) ( 0.20)(.80) 



700
 300

Critical point: z



0.10
 3.118
0.03207
= 3.09
0.001
H 0 may be rejected at any common level of significance.
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-10
BUSINESS STATISTICS
5th edi tion
Example 8-9: Carrying Out the Test
Standard Normal Distribution
0.4
f(z)
0.3
0.2
0.1
0.0
0
Nonrejection
Region
z
z0.001=3.09
Rejection
Region
Test Statistic=3.118
McGraw-Hill/Irwin
Since the value of the test
statistic is above the critical
point, even for a level of
significance as small as 0.001,
the null hypothesis may be
rejected, and we may conclude
that the proportion of
customers buying at least
$2500 of travelers checks is at
least 10% higher when
sweepstakes are on.
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
8-11
5th edi tion
Example 8-9: Using the Template
P-value =
0.0009, so
reject H0 at
the 5%
significance
level.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-12
BUSINESS STATISTICS
5th edi tion
Confidence Intervals for the Difference
between Two Population Proportions
A (1-) 100% large-sample confidence interval for the difference
between two population proportions:
( p1  p 2 )  z

 p (1  p )
1
 1
 n1 

p (1  p ) 
2
2 
n2


2
A 95% confidence interval using the data in example 8-9:
 p1 (1  p1 ) p 2 (1  p 2 ) 

  ( 0.4  0.2)  1.96 ( 0.4 )( 0.6)  ( 0.2)( 0.8)
( p1  p 2 )  z

n2
300
700

 n1

2
 0.2  (1.96)( 0.0321)  0.2  0.063  [ 0.137 ,0.263]
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
8-13
5th edi tion
Confidence Intervals for the Difference
between Two Population Proportions –
Using the Template – Using the Data
from Example 8-9
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-14
BUSINESS STATISTICS
5th edi tion
8-6 The F Distribution and a Test for
Equality of Two Population Variances
The F distribution is the distribution of the ratio of two chi-square random variables
that are independent of each other, each of which is divided by its own degrees of
freedom.
An F random variable with k1 and k2 degrees of freedom:
 12
F k ,k  
1
2
k1
 22
k2
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-15
BUSINESS STATISTICS
5th edi tion
The F Distribution
McGraw-Hill/Irwin
F Distributions with different Degrees of Freedom
f(F)
• The F random variable cannot
be negative, so it is bound by
zero on the left.
• The F distribution is skewed to
the right.
• The F distribution is identified
the number of degrees of
freedom in the numerator, k1,
and the number of degrees of
freedom in the denominator,
k2 .
F(25,30)
1.0
F(10,15)
0.5
F(5,6)
0.0
Aczel/Sounderpandian
0
1
2
3
4
5
F
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-16
BUSINESS STATISTICS
5th edi tion
Using the Table of the F Distribution
Critical Points of the F Distribution Cutting Off a
Right-Tail Area of 0.05
1
2
3
4
5
6
7
8
9
k2
1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5
2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38
3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81
4
7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00
5
6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77
6
5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10
7
5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68
8
5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39
9
5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18
10
4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02
11
4.84 3.98 3.59 3.36 3.20 3.09 3.01
3.01 2.95 2.90
12
4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80
13
4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71
14
4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65
15
4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59
0.7
0.6
0.5
f(F)
k1
F Distribution with 7 and 11 Degrees of Freedom
0.4
0.3
0.2
0.1
F
0.0
0
1
2
3
4
5
F0.05=3.01
The left-hand critical point to go along with F(k1,k2) is given by:
1
F k 2 ,k 1
Where F(k1,k2) is the right-hand critical point for an F random variable with the reverse
number of degrees of freedom.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-17
BUSINESS STATISTICS
5th edi tion
Critical Points of the F Distribution:
F(6, 9),  = 0.10
F Distribution with 6 and 9 Degrees of Freedom
0.7
0.05
0.90
0.6
The right-hand critical point read
directly from the table of the F
distribution is:
f(F)
0.5
F(6,9) =3.37
0.4
0.3
0.05
0.2
0.1
0.0
0
1
F0.95=(1/4.10)=0.2439
McGraw-Hill/Irwin
2
3
4
5
F
The corresponding left-hand critical
point is given by:
1
1

 0.2439
F 9 , 6 410
.
F0.05=3.37
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-18
BUSINESS STATISTICS
5th edi tion
Test Statistic for the Equality of Two
Population Variances
Test statistic for the equality of the variances of two normally
distributed populations:
F n 1,n 1
1
2
s12
 2
s2
 I: Two-Tailed Test
1 = 2
• H0: 1 = 2
• H1: 2
 II: One-Tailed Test
•
•
12
• H0: 1  2
• H1: 1  2
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-19
BUSINESS STATISTICS
5th edi tion
Example 8-10
The economist wants to test whether or not the event (interceptions and prosecution of insider traders) has
decreased the variance of prices of stocks.
Population1 : Before
n = 25
1
s 2  9.3
1
Population 2 : After
n = 24
2
s 2  3.0
2
  0.05
F
24,23
 2.01
  0.01
F
24,23
McGraw-Hill/Irwin
H 0: 
H1: 
2
1
2
1


2
2
21
2
2
s2
9.3
1
F
 F


 3.1
3.0
n1  1, n 2  1
24,23
s2
2




H 0 may be rejected at a 1% level of significance.
 2.70
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-20
BUSINESS STATISTICS
5th edi tion
Example 8-10: Solution
Distribution with 24 and 23 Degrees of Freedom
0.7
0.6
f(F)
0.5
0.4
0.3
0.2
0.1
F
0.0
0
1
2
F0.01=2.7
McGraw-Hill/Irwin
3
4
5
Test Statistic=3.1
Since the value of the test
statistic is above the critical
point, even for a level of
significance as small as 0.01,
the null hypothesis may be
rejected, and we may conclude
that the variance of stock
prices is reduced after the
interception and prosecution
of inside traders.
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
8-21
5th edi tion
Example 8-10: Solution Using the
Template
Observe that the pvalue for the test is
0.0042 which is less
than 0.01. Thus the
null hypothesis
must be rejected at
this level of
significance of 0.01.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
8-22
5th edi tion
Example 8-11: Testing the Equality of
Variances for Example 8-5
Population 1 Population 2
n = 14
1
2
2
s  0.12
1
  0.05
F
13,8
 3.28
  0.10
F
13,8
 2.50
McGraw-Hill/Irwin
n =9
2
2
2
s  0.11
2
2
2
H :  
0 1
2
2
2
H :  
1 1
2
s2
0.12 2
1
F
F


 119
.
2
2
n
1

1
,
n
2

1
13
,
8

   s 0.11
2
H may not be rejected at the 10% level of significance.
0
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
8-23
BUSINESS STATISTICS
5th edi tion
Example 8-11: Solution
F Distribution with 13 and 8 Degrees of Freedom
0.7
0.10
0.80
0.6
f(F)
0.5
0.4
0.3
0.10
0.2
0.1
0.0
0
1
F0.90=(1/2.20)=0.4545
Test Statistic=1.19
McGraw-Hill/Irwin
2
3
4
F0.10=3.28
5
F
Since the value of the test
statistic is between the critical
points, even for a 20% level of
significance, we can not reject
the null hypothesis. We
conclude the two population
variances are equal.
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
8-24
5th edi tion
Template to test for the Difference
between Two Population Variances:
Example 8-11
Observe that the pvalue for the test is
0.8304 which is larger
than 0.05. Thus the
null hypothesis
cannot be rejected at
this level of
significance of 0.05.
That is, one can
assume equal
variance.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
8-25
5th edi tion
The F Distribution Template to
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
8-26
5th edi tion
The Template for Testing Equality of
Variances
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
Penutup
• Pembandingan Dua Populasi merupakan
bagian dari pengujian Hipotesis dimana
populasinya lebih dari satu,hal ini juga
merupakan salah satu bentuk inferensial
statistik yang berupa pengambilan
kesimpulan/ pengambilan keputusan
tentang menolak atau tidak menolak
(menerima) suatu pernyataan/hipotesis
27