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
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Illustrasi dua kelompok
Dua Kelompok saling bebas
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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 (c) 2013 by Ali Muhson
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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)
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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 (c) 2013 by Ali Muhson
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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 1SD12 n2 1SD22 n1 n2 2
Nilai kritis t(; n1+n2-2) (c) 2013 by Ali Muhson
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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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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
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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)
(c) 2013 by Ali Muhson
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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? (c) 2013 by Ali Muhson
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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
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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 22
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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.
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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. (c) 2013 by Ali Muhson
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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 (c) 2013 by Ali Muhson
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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.
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Home Work Halaman 446 soal nomor 16 Halaman 458 soal nomor 42
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