05 Uji Beda Dua Kelompok 2013
3/27/2013
Ali Muhson, M.Pd.
Jenisnya
Uji Beda Rata-rata
Uji z
Uji t
Uji Beda Proporsi
Uji z
(c) 2013 by Ali Muhson
2
1
3/27/2013
Jenis Uji Beda Rata-rata dua
kelompok
Dua Kelompok Saling Bebas (Independent
Samples):
Uji z untuk uji populasi
Uji t untuk sampel kecil:
Pooled t test
Separate t test
Dua kelompok berpasangan (Paired samples)
Paired samples t test
(c) 2013 by Ali Muhson
3
Illustrasi dua kelompok
Dua Kelompok saling bebas
(c) 2013 by Ali Muhson
Dua Kelompok Berpasangan
4
2
3/27/2013
Jenis uji dua kelompok
saling bebas
Jumlah sampel
melebihi 30
Yes
Gunakan uji z.
No
Gunakan statistik
Nonparametrik
No
Apakah variansnya
homogen?
No
Apakah data berdistribusi
normal?
Yes
Apakah varians populasi
diketahui?
Yes
Yes
No
Gunakan uji z.
Gunakan separate t-test.
Gunakan
Pooled t test
(c) 2013 by Ali Muhson
5
Uji z Dua Kelompok Saling
Bebas
Tujuan Menguji perbedaan rerata dua
kelompok populasi yang saling bebas.
Syarat:
Sampel harus diambil secara random
Data berskala interval
Data pada masing-masing kelompok
berdistribusi normal
Dua kelompok tersebut tidak saling
berhubungan
Varians populasi diketahui, atau sampel
minimal 30
(c) 2013 by Ali Muhson
6
3
3/27/2013
Contoh Masalah
Apakah ada perbedaan tinggi
badan antara mahasiswa pria
dan wanita?
H0: μ1 = μ2
Ha: μ1 μ2
Benarkah bahwa hasil ujian
siswa kelas A lebih baik
daripada kelas B?
Apakah benar bahwa mobil
dengan sistem injeksi lebih
hemat BBM daripada yang tidak
menggunakan sistem injeksi?
H0: μ1 μ2
Ha: μ1 > μ2
H0: μ1 μ2
Ha: μ1 < μ2
(c) 2013 by Ali Muhson
7
Uji z dua kelompok
Rumus
z
X1 X 2
12 22
n1 n2
(c) 2013 by Ali Muhson
Nilai Kritis (z tabel)
Z (1-)
Standar Error
Perbedaan Rerata
( X 1 X 2 )
8
4
3/27/2013
Estimasi Parameter
Dengan tingkat keyakinan (1-) tertentu
dapat estimasi nilai parameter.
X
1
X 2 z1 SE 1 2 X 1 X 2 z1 SE
Margin Error
(ME)
(c) 2013 by Ali Muhson
9
Pooled t test
Tujuan Menguji perbedaan rerata dua kelompok
populasi yang saling bebas.
Syarat:
Sampel harus diambil secara random
Data berskala interval
Data pada masing-masing kelompok berdistribusi
normal
Dua kelompok tersebut tidak saling berhubungan
Varians populasi tidak diketahui, atau sampel
kurang dari 30
Varians kedua kelompok bersifat homogen
(c) 2013 by Ali Muhson
10
5
3/27/2013
Pooled t test
t
X1 X 2
1 1
Sp
n1 n2
Sp
Standar Error
Perbedaan Rerata
n1 1SD12 n2 1SD22
(S X 1 X 2 )
n1 n2 2
Nilai kritis t(; n1+n2-2)
(c) 2013 by Ali Muhson
11
Estimasi Parameter
Dengan tingkat keyakinan (1-) tertentu
dapat estimasi nilai parameter.
X
1
X 2 t SE 1 2 X 1 X 2 t SE
Margin Error
(ME)
(c) 2013 by Ali Muhson
12
6
3/27/2013
Separate t test
Tujuan Menguji perbedaan rerata dua kelompok
populasi yang saling bebas.
Syarat:
Sampel harus diambil secara random
Data berskala interval
Data pada masing-masing kelompok berdistribusi
normal
Dua kelompok tersebut tidak saling berhubungan
Varians populasi tidak diketahui, atau sampel
kurang dari 30
Varians kedua kelompok bersifat tidak homogen
(c) 2013 by Ali Muhson
Separate t test
t
X1 X 2
SD12 SD22
n1
n2
t tabel
w1
w2
w1t1 w2t 2
w1 w2
SD12
n1
13
T1 :
Alpha ()
db = n1 - 1
T2 :
Alpha ()
db = n2 – 1
SD22
n2
(c) 2013 by Ali Muhson
14
7
3/27/2013
Alternatif menghitung t
tabel
Dengan menghitung rumus db sebagai
berikut:
(c) 2013 by Ali Muhson
15
Estimasi Parameter
Dengan tingkat keyakinan (1-) tertentu
dapat estimasi nilai parameter.
X
1
X 2 t SE 1 2 X 1 X 2 t SE
Margin Error
(ME)
(c) 2013 by Ali Muhson
16
8
3/27/2013
Uji Homogenitas Varians
SD 2B
F
SD 2K
F tabel:
Alpha ()
db1 = nb – 1
db2 = nk – 1
Biasa ditulis F(; nb–1;
nk–1)
(c) 2013 by Ali Muhson
17
Contoh Soal
Seorang guru Ekonomi di SMA menyatakan
bahwa siswa kelas akselerasi memiliki nilai
yang lebih tinggi dibandingkan dengan kelas
reguler. Guna membuktikan pernyataan
tersebut dilakukan penelitian dengan
mengambil sampel secara acak di kedua kelas
tersebut. Dari 20 sampel siswa kelas
akselerasi diperoleh nilai rerata 82,4 dengan
standar deviasi 5,3 sementara 25 sampel
siswa di kelas reguler diperoleh nilai rerata
76,8 dengan standar deviasi 8,1. Dengan =
0,05, benarkah pernyataan guru tersebut?
(c) 2013 by Ali Muhson
18
9
3/27/2013
Contoh Hasil Analisis
IPK
Kelas
A
B
N
Mean
3.0871
2.7760
7
10
Std.
Std. Error
Deviation
Mean
.28459
.10756
.29463
.09317
(c) 2013 by Ali Muhson
19
Contoh Hasil Analisis
Levene's Test
for Equality of
Variances
t-test for Equality of Means
95% Confidence
Interval of the
Difference
F
IPK Equal
variances
assumed
Equal
variances not
assumed
Sig.
.305
.589
t
Sig. (2Mean
Std. Error
tailed) Difference Difference Lower
df
2.172
Upper
15
.046
.3111
.14324
.00584
.61644
2.186 13.365
.047
.3111
.14230
.00456
.61772
(c) 2013 by Ali Muhson
20
10
3/27/2013
Paired t test
Tujuan Menguji perbedaan rerata dua
kelompok populasi yang berpasangan.
Syarat:
Sampel harus diambil secara random
Data berskala interval
Data pada masing-masing kelompok
berdistribusi normal
Dua kelompok tersebut saling berhubungan
atau berpasangan
(c) 2013 by Ali Muhson
21
Paired t test
d
t
SDd
n
(c) 2013 by Ali Muhson
d X1 X 2
d
d
n
d X1 X 2
Nilai Kritis t:
Alpha ()
db = n - 1
22
11
3/27/2013
Estimasi Parameter
Dengan tingkat keyakinan (1-) tertentu
dapat estimasi nilai parameter.
X
1
X 2 t SE 1 2 X 1 X 2 t SE
Margin Error
(ME)
(c) 2013 by Ali Muhson
23
Paired t test
Example:
A reading center claims that students will perform better on a standardized reading
test after going through the reading course offered by their center. The table
shows the reading scores of 6 students before and after the course. At = 0.05,
is there enough evidence to conclude that the students’ scores after the course
are better than the scores before the course?
Student
1
3
4
5
Score (before)
85 96 70
76
81 78
Score (after)
88 85 89
86
92 89
H0: d 0
Ha: d > 0 (Claim)
2
6
Continued.
12
3/27/2013
t-Test for the Difference Between Means
H0: d 0
= 0.05
d.f. = 6 – 1 = 5
Ha: d > 0 (Claim)
d = (score before) – (score after)
Student
Score (before)
Score (after)
d
d2
-3
-2
-1
0
1
2
t
3
t0 = 2.015
1
2
3
4
5
6
85 96 70 76 81 78
88 85 89 86 92 89
3 11 19 10 11 11 d 43
9 121 361 100 121 121 d 2 833
d d 43 7.167
6
n
2
2
6(833) 1849
104.967 10.245
sd n( d ) ( d )
6(5)
n(n 1)
Continued.
t-Test for the Difference Between
Means
H0: d 0
Ha: d > 0 (Claim)
The standardized test statistic is
d μd 7.167 0
t
1.714.
sd n 10.245 6
-3
-2
-1
0
1
2
3
t
t0 = 2.015
Fail to reject H0.
There is not enough evidence at the 5% level to support the claim that the
students’ scores after the course are better than the scores before the course.
(c) 2013 by Ali Muhson
26
13
3/27/2013
Hasil Analisis:
Mean
Pair 1
Std.
Deviation
N
Std. Error
Mean
Nilai Pre Test
54.4000
10
7.33636
2.31996
Nilai Post Test
68.7000
10
8.75658
2.76908
Mean
Pair Nilai Pre
1
Test - Nilai
Post Test
Paired Differences
95% Confidence
Interval of the
Std.
Std.
Difference
Deviatio Error
n
Mean
Lower
Upper
-14.300 8.94489 2.82862
-20.699
t
-7.9012 -5.055
df
9
Sig. (2tailed)
.001
(c) 2013 by Ali Muhson
27
Uji Beda Proporsi
Tujuan menguji perbedaan proporsi dua
kelompok
Syarat:
Sampel harus diambil secara random
Dua kelompok tersebut saling bebas
Jumlah sampel harus cukup besar untuk dapat
menggunakan distribusi normal, yaitu:
n1p1 5, n1q1 5,
n2p2 5, dan n2q2 5.
(c) 2013 by Ali Muhson
28
14
3/27/2013
Uji Beda Proporsi
z
p
p1 p2
1 1
pq
n1 n2
X1 X 2
n1 n2
p1
p2
X1
n1
X2
n2
q 1 p
(c) 2013 by Ali Muhson
29
Example
A recent survey stated that male college
students smoke less than female college
students. In a survey of 1245 male students,
361 said they smoke at least one pack of
cigarettes a day. In a survey of 1065 female
students, 341 said they smoke at least one
pack a day. At = 0,01, can you support the
claim that the proportion of male college
students who smoke at least one pack of
cigarettes a day is lower then the proportion
of female college students who smoke at least
one pack a day?
(c) 2013 by Ali Muhson
30
15
3/27/2013
Two Sample z-Test for
Proportions
H0: p1 p2
z0 = 2.33
-3
Ha: p1 < p2 (Claim)
p
-2 -1
0
1
2
z
3
x1 x 2 n1 pˆ1 n2 pˆ2
361 341
702
0.304
n1 n2
n1 n2
1245 1065 2310
q 1 p 1 0.304 0.696
Because 1245(0.304), 1245(0.696), 1065(0.304), and 1065(0.696)
are all at least 5, we can use a two-sample z-test.
Continued.
(c) 2013 by Ali Muhson
31
Two Sample z-Test for
Proportions
H0: p1 p2
z0 = 2.33
-3
Ha: p1 < p2 (Claim)
z
p1 p 2
1
1
pq
n1 n 2
-2 -1
0
1
2
3
z
(0.29 0.32) 0
1.56
1
1
(0.304)(0.696)
1245 1065
Fail to reject H0.
There is not enough evidence at the 1% level to support the claim that the
proportion of male college students who smoke is lower then the proportion of
female college students who smoke.
(c) 2013 by Ali Muhson
32
16
3/27/2013
Home Work
Halaman 446 soal nomor 16
Halaman 458 soal nomor 42
(c) 2013 by Ali Muhson
33
17
Ali Muhson, M.Pd.
Jenisnya
Uji Beda Rata-rata
Uji z
Uji t
Uji Beda Proporsi
Uji z
(c) 2013 by Ali Muhson
2
1
3/27/2013
Jenis Uji Beda Rata-rata dua
kelompok
Dua Kelompok Saling Bebas (Independent
Samples):
Uji z untuk uji populasi
Uji t untuk sampel kecil:
Pooled t test
Separate t test
Dua kelompok berpasangan (Paired samples)
Paired samples t test
(c) 2013 by Ali Muhson
3
Illustrasi dua kelompok
Dua Kelompok saling bebas
(c) 2013 by Ali Muhson
Dua Kelompok Berpasangan
4
2
3/27/2013
Jenis uji dua kelompok
saling bebas
Jumlah sampel
melebihi 30
Yes
Gunakan uji z.
No
Gunakan statistik
Nonparametrik
No
Apakah variansnya
homogen?
No
Apakah data berdistribusi
normal?
Yes
Apakah varians populasi
diketahui?
Yes
Yes
No
Gunakan uji z.
Gunakan separate t-test.
Gunakan
Pooled t test
(c) 2013 by Ali Muhson
5
Uji z Dua Kelompok Saling
Bebas
Tujuan Menguji perbedaan rerata dua
kelompok populasi yang saling bebas.
Syarat:
Sampel harus diambil secara random
Data berskala interval
Data pada masing-masing kelompok
berdistribusi normal
Dua kelompok tersebut tidak saling
berhubungan
Varians populasi diketahui, atau sampel
minimal 30
(c) 2013 by Ali Muhson
6
3
3/27/2013
Contoh Masalah
Apakah ada perbedaan tinggi
badan antara mahasiswa pria
dan wanita?
H0: μ1 = μ2
Ha: μ1 μ2
Benarkah bahwa hasil ujian
siswa kelas A lebih baik
daripada kelas B?
Apakah benar bahwa mobil
dengan sistem injeksi lebih
hemat BBM daripada yang tidak
menggunakan sistem injeksi?
H0: μ1 μ2
Ha: μ1 > μ2
H0: μ1 μ2
Ha: μ1 < μ2
(c) 2013 by Ali Muhson
7
Uji z dua kelompok
Rumus
z
X1 X 2
12 22
n1 n2
(c) 2013 by Ali Muhson
Nilai Kritis (z tabel)
Z (1-)
Standar Error
Perbedaan Rerata
( X 1 X 2 )
8
4
3/27/2013
Estimasi Parameter
Dengan tingkat keyakinan (1-) tertentu
dapat estimasi nilai parameter.
X
1
X 2 z1 SE 1 2 X 1 X 2 z1 SE
Margin Error
(ME)
(c) 2013 by Ali Muhson
9
Pooled t test
Tujuan Menguji perbedaan rerata dua kelompok
populasi yang saling bebas.
Syarat:
Sampel harus diambil secara random
Data berskala interval
Data pada masing-masing kelompok berdistribusi
normal
Dua kelompok tersebut tidak saling berhubungan
Varians populasi tidak diketahui, atau sampel
kurang dari 30
Varians kedua kelompok bersifat homogen
(c) 2013 by Ali Muhson
10
5
3/27/2013
Pooled t test
t
X1 X 2
1 1
Sp
n1 n2
Sp
Standar Error
Perbedaan Rerata
n1 1SD12 n2 1SD22
(S X 1 X 2 )
n1 n2 2
Nilai kritis t(; n1+n2-2)
(c) 2013 by Ali Muhson
11
Estimasi Parameter
Dengan tingkat keyakinan (1-) tertentu
dapat estimasi nilai parameter.
X
1
X 2 t SE 1 2 X 1 X 2 t SE
Margin Error
(ME)
(c) 2013 by Ali Muhson
12
6
3/27/2013
Separate t test
Tujuan Menguji perbedaan rerata dua kelompok
populasi yang saling bebas.
Syarat:
Sampel harus diambil secara random
Data berskala interval
Data pada masing-masing kelompok berdistribusi
normal
Dua kelompok tersebut tidak saling berhubungan
Varians populasi tidak diketahui, atau sampel
kurang dari 30
Varians kedua kelompok bersifat tidak homogen
(c) 2013 by Ali Muhson
Separate t test
t
X1 X 2
SD12 SD22
n1
n2
t tabel
w1
w2
w1t1 w2t 2
w1 w2
SD12
n1
13
T1 :
Alpha ()
db = n1 - 1
T2 :
Alpha ()
db = n2 – 1
SD22
n2
(c) 2013 by Ali Muhson
14
7
3/27/2013
Alternatif menghitung t
tabel
Dengan menghitung rumus db sebagai
berikut:
(c) 2013 by Ali Muhson
15
Estimasi Parameter
Dengan tingkat keyakinan (1-) tertentu
dapat estimasi nilai parameter.
X
1
X 2 t SE 1 2 X 1 X 2 t SE
Margin Error
(ME)
(c) 2013 by Ali Muhson
16
8
3/27/2013
Uji Homogenitas Varians
SD 2B
F
SD 2K
F tabel:
Alpha ()
db1 = nb – 1
db2 = nk – 1
Biasa ditulis F(; nb–1;
nk–1)
(c) 2013 by Ali Muhson
17
Contoh Soal
Seorang guru Ekonomi di SMA menyatakan
bahwa siswa kelas akselerasi memiliki nilai
yang lebih tinggi dibandingkan dengan kelas
reguler. Guna membuktikan pernyataan
tersebut dilakukan penelitian dengan
mengambil sampel secara acak di kedua kelas
tersebut. Dari 20 sampel siswa kelas
akselerasi diperoleh nilai rerata 82,4 dengan
standar deviasi 5,3 sementara 25 sampel
siswa di kelas reguler diperoleh nilai rerata
76,8 dengan standar deviasi 8,1. Dengan =
0,05, benarkah pernyataan guru tersebut?
(c) 2013 by Ali Muhson
18
9
3/27/2013
Contoh Hasil Analisis
IPK
Kelas
A
B
N
Mean
3.0871
2.7760
7
10
Std.
Std. Error
Deviation
Mean
.28459
.10756
.29463
.09317
(c) 2013 by Ali Muhson
19
Contoh Hasil Analisis
Levene's Test
for Equality of
Variances
t-test for Equality of Means
95% Confidence
Interval of the
Difference
F
IPK Equal
variances
assumed
Equal
variances not
assumed
Sig.
.305
.589
t
Sig. (2Mean
Std. Error
tailed) Difference Difference Lower
df
2.172
Upper
15
.046
.3111
.14324
.00584
.61644
2.186 13.365
.047
.3111
.14230
.00456
.61772
(c) 2013 by Ali Muhson
20
10
3/27/2013
Paired t test
Tujuan Menguji perbedaan rerata dua
kelompok populasi yang berpasangan.
Syarat:
Sampel harus diambil secara random
Data berskala interval
Data pada masing-masing kelompok
berdistribusi normal
Dua kelompok tersebut saling berhubungan
atau berpasangan
(c) 2013 by Ali Muhson
21
Paired t test
d
t
SDd
n
(c) 2013 by Ali Muhson
d X1 X 2
d
d
n
d X1 X 2
Nilai Kritis t:
Alpha ()
db = n - 1
22
11
3/27/2013
Estimasi Parameter
Dengan tingkat keyakinan (1-) tertentu
dapat estimasi nilai parameter.
X
1
X 2 t SE 1 2 X 1 X 2 t SE
Margin Error
(ME)
(c) 2013 by Ali Muhson
23
Paired t test
Example:
A reading center claims that students will perform better on a standardized reading
test after going through the reading course offered by their center. The table
shows the reading scores of 6 students before and after the course. At = 0.05,
is there enough evidence to conclude that the students’ scores after the course
are better than the scores before the course?
Student
1
3
4
5
Score (before)
85 96 70
76
81 78
Score (after)
88 85 89
86
92 89
H0: d 0
Ha: d > 0 (Claim)
2
6
Continued.
12
3/27/2013
t-Test for the Difference Between Means
H0: d 0
= 0.05
d.f. = 6 – 1 = 5
Ha: d > 0 (Claim)
d = (score before) – (score after)
Student
Score (before)
Score (after)
d
d2
-3
-2
-1
0
1
2
t
3
t0 = 2.015
1
2
3
4
5
6
85 96 70 76 81 78
88 85 89 86 92 89
3 11 19 10 11 11 d 43
9 121 361 100 121 121 d 2 833
d d 43 7.167
6
n
2
2
6(833) 1849
104.967 10.245
sd n( d ) ( d )
6(5)
n(n 1)
Continued.
t-Test for the Difference Between
Means
H0: d 0
Ha: d > 0 (Claim)
The standardized test statistic is
d μd 7.167 0
t
1.714.
sd n 10.245 6
-3
-2
-1
0
1
2
3
t
t0 = 2.015
Fail to reject H0.
There is not enough evidence at the 5% level to support the claim that the
students’ scores after the course are better than the scores before the course.
(c) 2013 by Ali Muhson
26
13
3/27/2013
Hasil Analisis:
Mean
Pair 1
Std.
Deviation
N
Std. Error
Mean
Nilai Pre Test
54.4000
10
7.33636
2.31996
Nilai Post Test
68.7000
10
8.75658
2.76908
Mean
Pair Nilai Pre
1
Test - Nilai
Post Test
Paired Differences
95% Confidence
Interval of the
Std.
Std.
Difference
Deviatio Error
n
Mean
Lower
Upper
-14.300 8.94489 2.82862
-20.699
t
-7.9012 -5.055
df
9
Sig. (2tailed)
.001
(c) 2013 by Ali Muhson
27
Uji Beda Proporsi
Tujuan menguji perbedaan proporsi dua
kelompok
Syarat:
Sampel harus diambil secara random
Dua kelompok tersebut saling bebas
Jumlah sampel harus cukup besar untuk dapat
menggunakan distribusi normal, yaitu:
n1p1 5, n1q1 5,
n2p2 5, dan n2q2 5.
(c) 2013 by Ali Muhson
28
14
3/27/2013
Uji Beda Proporsi
z
p
p1 p2
1 1
pq
n1 n2
X1 X 2
n1 n2
p1
p2
X1
n1
X2
n2
q 1 p
(c) 2013 by Ali Muhson
29
Example
A recent survey stated that male college
students smoke less than female college
students. In a survey of 1245 male students,
361 said they smoke at least one pack of
cigarettes a day. In a survey of 1065 female
students, 341 said they smoke at least one
pack a day. At = 0,01, can you support the
claim that the proportion of male college
students who smoke at least one pack of
cigarettes a day is lower then the proportion
of female college students who smoke at least
one pack a day?
(c) 2013 by Ali Muhson
30
15
3/27/2013
Two Sample z-Test for
Proportions
H0: p1 p2
z0 = 2.33
-3
Ha: p1 < p2 (Claim)
p
-2 -1
0
1
2
z
3
x1 x 2 n1 pˆ1 n2 pˆ2
361 341
702
0.304
n1 n2
n1 n2
1245 1065 2310
q 1 p 1 0.304 0.696
Because 1245(0.304), 1245(0.696), 1065(0.304), and 1065(0.696)
are all at least 5, we can use a two-sample z-test.
Continued.
(c) 2013 by Ali Muhson
31
Two Sample z-Test for
Proportions
H0: p1 p2
z0 = 2.33
-3
Ha: p1 < p2 (Claim)
z
p1 p 2
1
1
pq
n1 n 2
-2 -1
0
1
2
3
z
(0.29 0.32) 0
1.56
1
1
(0.304)(0.696)
1245 1065
Fail to reject H0.
There is not enough evidence at the 1% level to support the claim that the
proportion of male college students who smoke is lower then the proportion of
female college students who smoke.
(c) 2013 by Ali Muhson
32
16
3/27/2013
Home Work
Halaman 446 soal nomor 16
Halaman 458 soal nomor 42
(c) 2013 by Ali Muhson
33
17