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
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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
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Illustrasi dua kelompok
Dua Kelompok saling bebas
(c) 2013 by Ali Muhson
Dua Kelompok Berpasangan
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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
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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
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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?
H0: μ1  μ2
Ha: μ1 > μ2
 Apakah benar bahwa mobil
dengan sistem injeksi lebih
hemat BBM daripada yang tidak
menggunakan sistem injeksi?
H0: μ1  μ2
Ha: μ1 < μ2
(c) 2013 by Ali Muhson
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Uji z dua kelompok
Rumus
z
Nilai Kritis (z tabel)
X1  X 2
 12  22

n1 n2
(c) 2013 by Ali Muhson
 Z (1-)
Standar Error
Perbedaan Rerata
( X 1  X 2 )
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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
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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
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Pooled t test
t
X1  X 2
1 1
Sp

n1 n2
Sp 
Standar Error
Perbedaan Rerata
(S X 1  X 2 )
n1  1SD12  n2  1SD22
n1  n2  2
 Nilai kritis t(; n1+n2-2)
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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)
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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
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Separate t test
t
X1  X 2
SD12 SD22

n1
n2
t tabel 
w1t1  w2t 2
w1  w2
w1 
SD12
n1
w2 
SD22
n2
(c) 2013 by Ali Muhson
 T1 :
 Alpha ()
 db = n1 - 1
 T2 :
 Alpha ()
 db = n2 – 1
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Alternatif menghitung t
tabel
 Dengan menghitung rumus db sebagai
berikut:
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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)
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Uji Homogenitas Varians
SD 2B
F
SD 2K
 F tabel:
 Alpha ()
 db1 = nb – 1
 db2 = nk – 1
 Biasa ditulis F(; nb–1;
nk–1)
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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?
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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
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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
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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
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Paired t test
d  X1  X 2
d
t
SDd
n
(c) 2013 by Ali Muhson
d
d
n
d  X1  X 2
 Nilai Kritis t:
 Alpha ()
 db = n - 1
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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
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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.
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t-Test for the Difference Between Means
d.f. = 6 – 1 = 5
H0: d  0
 = 0.05
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
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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
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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.
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Uji Beda Proporsi
z
p
p1  p2
1 1
pq  
 n1 n2 
X1  X 2
n1  n2
p1 
X1
n1
p2 
X2
n2
q  1 p
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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
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Two Sample z-Test for
Proportions
H0: p1  p2
-3
p
-2 -1
z0 = 2.33
Ha: p1 < p2 (Claim)
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
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Two Sample z-Test for
Proportions
H0: p1  p2
-3
z0 = 2.33
Ha: p1 < p2 (Claim)
z
-2 -1
p1  p 2
 1
1 

pq  
 n1 n 2 

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
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Home Work
 Halaman 446 soal nomor 16
 Halaman 458 soal nomor 42
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