STATISTIK INFERENSI:
PENGUJIAN HIPOTESIS BAGI ANALISIS KORELASI
ATAU HUBUNGAN
(UJIAN – r )
Rohani Ahmad Tarmizi - EDU5950
1
STATISTIK INFERENSI ATAU PENTAKBIRAN
(Inferential Statistics)
 Bertujuan untuk menerangkan ciri
populasi berdasarkan data yang dikumpul
daripada sampel.
 Tujuan ini berkait rapat dengan objektif
kajian serta hipotesis atau soalan kajian.
 Membolehkan penyelidik membuat
kesimpulan bahawa terdapat “statistik
yang signifikan” atau “statistical
significance” yang bermaksud boleh
diterima pakai dengan meluas,
meyakinkan.
LANGKAH PENGUJIAN HIPOTESIS
 L1. Nyatakan hipotesis hipotesis statistik/sifar




(H0) dan hipotesis penyelidikan (HA) –
BERARAH ATAU TIDAK BERARAH
L2. Tetapkan aras signifikan, taburan
persampelan dan statistik pengujian yang akan
digunakan – ARAS ALPHA = 0.01/ 0.05/ 0.10,
TABURAN PERSAMPELAN z, t, F, r…
STATISTIK PENGUJIAN (z, t, F, r…)
L3. Tentukan nilai kritikal bagi taburan
persampelan yang akan digunakan RUJUK JADUAL z, t, F, r…
L4. Kirakan statistik pengujian (tests statistics)
bagi taburan persampelan tersebut – RUJUK
FORMULA
L5. Buat keputusan, tafsiran, dan kesimpulan.
L1. Nyatakan hipotesis
 Hipotesis penyelidikan –
Terdapat perbezaan yang signifikan antara
min tahap kepimpinan pengajaran Pengetua
di Sekolah berprestasi tinggi berbanding
dengan sekolah Swasta .
 Hipotesis nol/sifar –
Tiada terdapat perbezaan yang signifikan
antara min tahap kepimpinan pengajaran
Pengetua di sekolah berprestasi tinggi
berbanding dengan sekolah Swasta.
1. Nyata hipotesis nol dan penyelidikan.
H O : µ 1 = µ2
HA : µ1 ≠ µ2
H O : µ1 ≥ µ 2
H A : µ 1 < µ2
H O : µ1 ≤ µ 2
H A : µ 1 > µ2
L1. Nyatakan hipotesis (dua kumpulan)
 Hipotesis penyelidikan –
Terdapat perbezaan yang signifikan antara tahap
kepimpinan pengajaran Pengetua, GPK2 dan
GPK1.
 Hipotesis nol/sifar –
Tiada terdapat perbezaan yang signifikan antara
tahap kepimpinan pengajaran Pengetua, GPK2
dan GPK1.
ANOVA’S Hypothesis
H0: 1 = 2 = 3 = ... = c
•All population means are equal
•No treatment effect (NO variation in means among groups)
H1: not all the k are equal
•At least ONE population mean is different
(Others may be the same!)
•There is treatment effect
Does NOT mean that all the means are different:
1  2  ...  c
UJIAN PERBANDINGAN MIN
 Ujian-t bebas (independent sample t-test)
 diguna untuk menguji hipotesis bahawa tiada/ada
perbezaan min kemahiran IT antara kumpulan lelaki
dan perempuan.
 Ujian-t bersandar (paired sample t-test)
 digunakan untuk menguji hipotesis bahawa tiada/ada
perbezaan min tahap kepemimpinan mengarah
dengan min tahap kepemimpinan partisipatif.
 Ujian analisis varians (analysis of variance, F test)
 digunakan untuk menguji hipotesis tiada/ada
perbezaan min kepuasan bekerja antara kumpulan
yang berbeza taraf pekerjaan.
UJIAN PERBANDINGAN MIN-MIN
 Terdapat perbezaan min antara dua
kumpulan yang dikaji
(hipotesis penyelidikan)
 Tiada terdapat perbezaan min antara
kumpulan yang dikaji
(hipotesis sifar atau statistik).
 Disini yang dibanding adalah min ia itu skor
pembolehubah bersandar daripada dua atau
lebih daripada dua kumpulan yang dikaji.
Uji diri anda!!!-Apakah pengujian statistik yang
diperlukan, JIKA ANDA HENDAK?
 1: Mengkaji sama ada terdapat perkaitan min kecekapan




menjalankan penyelidikan dengan min pengetahuan
statistik di kalangan pelajar Masters.
2: Menentukan sama ada terdapat hubungan antara
tahap penglihatan “left field” dengan “right field” di
kalangan pelajar.
3: Menguji terdapat hubungan antara kemahiran
berkomunikasi dengan tahap keyakinan-diri
dikalangan pelajar FPP.
4: Menguji terdapat hubungan kemahiran IT antara
abang dan adik.
5. Mengenal pasti hubungan kecerdasan intelek dengan
kecerdasan emosi
BAGI TUJUAN SEDEMIKIAN!!
ANALISIS KORELASI
(Correlation Analysis)
 Analisis ini membolehkan penyelidik menguji
hipotesis bahawa terdapat
 hubungan (relationship),
 korelasi (correlation) atau
 perkaitan (association)
antara dua atau lebih pembolehubah
 Analisis ini bertujuan untuk menentukan
hubungan/korelasi antara pembolehubahpembolehubah yang dikaji yang diperoleh/diukur
daripada responden kajian iaitu kumpulan sampel
ataupun populasi.
 Analisis ini digunakan untuk menjawab persoalan
kajian seperti berikut:
 Adakah terdapat hubungan antara
dua pembolehubah tersebut?
 “Is there relationship between the two
variables?”
 Sejauh manakah hubungan tersebut?
 “How strong is the relationship?”
 Apakah arah hubungan tersebut?
 “What is the direction of the
relationship?”
ANALISIS KORELASI
 Analisis korelasi juga boleh dilanjutkan
menjadi beberapa pembolehubah –
MULTICORELATIONAL ANALYSIS
 Ia mengukur sejauh manakah dua atau
lebih pembolehubah berubah (covary)
secara serentak ataupun bersama-sama.
 It is a measure of how variables covary
together, hence the word CORRELATION
ANALISIS KORELASI
 Analisis juga membabitkan dua kategori
pembolehubah iaitu pembolehubah prediktif dan
pembolehubah kriterion.
 P/U prediktif adalah yang memberi kesan atau
mempengaruhi P/U yang kedua.
 P/U kriterion adalah yang menerima kesan atau
pengaruh daripada P/U pertama.
 X (prediktif)
Y (kriterion)
 X1, X2, X3,..
Y (kriterion)
 Walau bagaimanapun, analisis ini hanya memeri
gambaran hubungan dan tidak memberi rumusan
“cause-and-effect relationship”.
 Sebagai contoh, penyelidik hendak
menentukan hubungan antara:
 minat terhadap bidang dengan prestasi,
 pendapatan dan kepuasan bekerja
 kadar baja dan pertumbuhan pokok
 frekuensi merokok dengan frekuensi
mendapat serangan jantung
 Umur dengan kadar ingatan
 jam mentelaah, amalan pemakanan, IQ
dengan prestasi.
Other Examples of Correlation Analysis
 A researcher interested in the
relationship between measures of
depression and violence among
fathers.
 Relationship between age in years
and performance in motor skills.
 Relationship between hours of
leisure and performance in course.
Specific Example
For seven random
summer days, a
person recorded the
temperature and
their water
consumption,
during a three-hour
period spent
outside.
Temperature (F)
Water
Consumption
(ounces)
75
83
85
85
92
97
99
16
20
25
27
32
48
48
How would you describe the graph?
Other Direction of Correlation
Correlation
A relationship between two variables.
Explanatory
(Independent)
Variable
x
Hours of Training
Shoe Size
Cigarettes smoked per day
Score on MUET
Height
y
Response
(Dependent)
Variable
Number of Accidents
Height
Lung Capacity
Grade Point Average
IQ
What type of relationship exists between the two
variables and is the correlation significant?
Dua Cara Menentukan Korelasi
 Secara bergambar iaitu dinamakan
gambarajah sebaran (scatter diagram)
yang menunjukkan pola kedudukan
pasangan titik-titik.
 Daripada gambarajah sebaran kita
dapat merumus keteguhan
(magnitud) korelasi tersebut serta
arah korelasinya.
Dua Cara Menentukan Korelasi
 Secara berangka iaitu dengan
menentukan pekali, koefisi atau
indeks.
 Daripada pekali tersebut kita dapat
mengetahui keteguhan (magnitud)
korelasi tersebut serta arahnya sama
positif atau negatif.
MEMBINA GAMBAR RAJAH SEBARAN
 Lakarkan dua paksi mengufuk dan
mencancang
 Letakkan p/u X (IV or predictive) pada paksi
mengufuk dan tandakan paksi tersebut
 Letakkan p/u Y (DV or criterion) pada paksi
mencancang.
 Plotkan titik-kedudukan bagi setiap
pasangan skor di lakaran tersebut ia itu titik
persilangan titik bagi X dan titik bagi Y bagi
setiap pasangan skor.
 Pola Kedudukan Titik Serta Garis Lurus Yang
Terbentuk Menggambarkan Korelasi Atau
Hubungan Iaitu:
 titik-titik yang terletak di atas garis lurus
yang mendongak menunjukkan korelasi
sempurna dan positif.
 titik-titik yang terletak di atas garis lurus
yang menunduk menunjukkan korelasi
sempurna dan negatif.
 titik-titik yang bersepah tanpa pola garis
lurus menunjukkan tiada korelasi.
 Pola Kedudukan Titik Serta Garis Lurus Yang
Terbentuk Menggambarkan Korelasi Atau
Hubungan Iaitu:
 titik-titik yang menghampir dengan garis
lurus yang mendongak menunjukkan
korelasi teguh dan positif.
 titik-titik yang berjauhan daripada garis
lurus serta juga mendongak menunjukkan
korelasi lemah dan positif.
 Pola Kedudukan Titik Serta Garis Lurus Yang
Terbentuk Menggambarkan Korelasi Atau
Hubungan Iaitu:
 titik-titik yang menghampiri dengan garis
lurus yang menunduk menunjukkan
korelasi teguh dan negatif.
 titik-titik yang berjauhan daripada garis
lurus yang juga menunduk menunjukkan
korelasi lemah dan negatif.
Scatter Plots and Types of Correlation
x = SAT score
y = GPA
GPA
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
300
350
400
450
500
550
600
650
700
750
800
Math SAT
Positive Correlation
as x increases y increases
Scatter Plots and Types of Correlation
x = hours of training
y = number of accidents
Accidents
60
50
40
30
20
10
0
0
2
4
6
8
10
12
14
16
18
Hours of Training
Negative Correlation
as x increases, y decreases
20
x
Absences
x
8
2
5
12
15
9
6
Final
Grade
Grade
95
90
85
80
75
70
65
60
55
50
45
40
0
2
4
6
8
10
12
Absences
y
78
92
90
58
43
74
81
14
16
x
IQ
Scatter Plots and Types of Correlation
x = height
y = IQ
160
150
140
130
120
110
100
90
80
60
64
68
72
76
Height
No linear correlation
80
Displays of scores in a Scatterplot
Hours of
Internet Depression
use
scores
per week from 15-45
Laura
Chad
Patricia
Bill
Mary
Todd
Angela
David
Maxine
John
Mean Score
17
13
5
9
5
15
7
6
2
18
10
30
41
18
20
25
44
20
30
17
48
29.3
Depression scores
Y=D.V.
50
-
40
30
+
M
20
+
10
M
5
-
10 15 20
Hours of Internet Use
X=I.V.
Association Between Two Scores Linear and
non-linear patterns
A. Positive Linear (r=+.75)B. Negative Linear (r=-.68)
C. No
Correlation
(r=.00)
Linear and non-linear patterns
D. Curvilinear
E. Curvilinear
F. Curvilinear
Analisis Korelasi Menunjukkan
3 perkara penting, iaitu:
 Arah/Direction (positive or negative)
 Bentuk/Form (linear or non-linear)
 Kekuatan/Magnitude (size of coefficient)
PEKALI ATAU KOEFISI KORELASI
 TERDAPAT BEBERAPA JENIS PEKALI
KORELASI IAITU:
 Pearson product-moment correlation
 Digunakan apabila p/u x dan y adalah pada skala sela
atau nisbah atau gabungan kedua-duanya.
 Spearman rho correlation
 Digunakan apabila p/u x dan y adalah pada skala
ordinal atau gabungan ordinal dengan sela/nisbah.
 Point-biserial correlation
 Digunakan apabila p/u x adalah dikotomus dan p/u y
adalah pada skala sela atau nisbah.
Pekali Pearson
r =
n [xy] - [xy]
[ n  x2 - ( x) 2 ] [ n  y2 - ( y) 2 ]
n = bilangan pasangan skor
 x y = jumlah skor x didarab dengan skor y
 x = jumlah skor x
 y = jumlah skor y
Pekali Spearman
r =
1
-
[6B2]
n [ n2 - 1 ]
n = bilangan pasangan skor
 B = jumlah beza pasangan setiap skor
Pekali Point-biserial
r =
y1 – y2
[ n1 n2 ]
sy
n[n-1]
Interpreting Value of the Correlation
Coefficient
 The coefficient vary from zero to one.
 Zero means there is simply no correlation between
the two variables.
 One means a perfect correlation.
 Values approaching one is considered highly
correlated.
 Values approaching zero is least correlated.
 A middle score indicates moderate or average
correlation.
Interpreting Positive
Correlation
 A correlation can either be positively or negatively
correlated.
 Positive correlation is indicated by a positive value
is obtained.
 This means that the two variables are varying from
each other: as X increases Y will also increase.
 Y being the criterion variables (dependent
variables) and X the predictive variables
(independent variables)
Non-linear associations statistics
 Spearman rho (rs) - correlation
coefficient for nonlinear ordinal data
 Point-biserial - used to correlate
continuous interval data with a
dichotomous variable
 Phi-coefficient - used to determine the
degree of association when both
variable measures are dichotomous
Association Between Two Scores Degree and
strength of association
 .20–.35:
 When correlations range from .20 to .35, there is only a






slight relationship
.35–.65:
When correlations are above .35, they are useful for
limited prediction.
.66–.85:
When correlations fall into this range, good prediction
can result from one variable to the other. Coefficients
in this range would be considered very good.
.86 and above:
Correlations in this range are typically achieved for
studies of construct validity or test-retest reliability.
Interpreting Value of the Correlation
Coefficient
 The coefficient vary from zero to one.
 Zero means there is simply no correlation between
the two variables.
 One means a perfect correlation.
 Values approaching one is considered highly
correlated.
 Values approaching zero is least correlated.
 A middle score indicates moderate or average
correlation.
Guildford Rule of Thumb
r
Strength of Relationship
< 0.2
Negligible Relationship
0.2 – 0.4
Low Relationship
0.4 – 0.7
Moderate Relationship
0.7 – 0.9
High Relationship
> 0.9
Very high Relationship
Other Strengths of AssociationBy Johnson and Nelson (1986)
r-value
Interpretation
0.00
No relationship
0.01-0.19
Low relationship
0.20-0.49
Slightly Moderate relationship
0.50-0.69
Moderate relationship
0.70-0.99
Strong relationship
1.00
Perfect relationship
TYPES OF CORRELATION
 Pearson correlation coefficient.
 Spearman’s rank correlation coefficient.
 Point-biserial correlation coefficient.
L1. Nyatakan hipotesis
 Hipotesis penyelidikan –
Terdapat hubungan yang signifikan antara tahap
kepimpinan pengajaran Pengetua dengan
prestasi akademik sekolah di Sabah
 Hipotesis nol/sifar –
Tiada terdapat hubungan yang signifikan antara
tahap kepimpinan pengajaran Pengetua dengan
prestasi akademik sekolah di Sabah
L2. TETAPKAN ARAS ALPHA = 0.01/ 0.05/ 0.10,
TABURAN PERSAMPELAN, STATISTIK PENGUJIAN
 Nilai alpha ditetapkan oleh penyelidik.
 Ia merupakan nilai penetapan bahawa penyelidik akan
menerima sebarang ralat semasa membuat keputusan
pengujian hipotesis tersebut.
 Ralat yang sekecil-kecilnya ialah 0.01 (1%), 0.05 (5%)
atau 0.10(10%).
 Nilai ini juga dipanggil nilai signifikan, aras signifikan,
atau aras alpha.
L2. Taburan Persampelan
 Taburan yang bersesuaian dengan analisis yang
dijalankan. Ia merupakan model taburan
korelasi yang mana nilai korelasi itu bertabur
secara normal.
 Di kawasan kritikal terletak nilai korelasi yang
“luar biasa” -> Ha adalah benar
 Dikawasan tak kritikal terletak nilai korelasi
yang “biasa” -> Ho adalah benar
L3. Nilai Kritikal
 Nilai kritikal adalah nilai yang menjadi sempadan
bagi kawasan Ho benar dan Hp benar.
 Nilai ini merupakan nilai dimana penyelidik
meletakkan penetapan sama ada cukup bukti
untuk menolak Ho (maka boleh menerima Hp)
ataupun tidak cukup bukti menolak Ho
(menerima Ho).
 Nilai ini bergantung kepada nilai alpha dan arah
pengujian hipotesis yang dilakukan.
L4. Nilai Statistik Pengujian
 Ini adalah nilai yang dikira dan dijadikan bukti
sama ada hipotesis sifar benar atau salah.
 Jika nilai statistik pengujian masuk dalam kawasan
kritikal maka Ho adalah salah, ditolak dan Hp
diterima
 Jika nilai statistik pengujian masuk dalam kawasan
tak kritikal maka Ho adalah benar, maka terima
Ho.
L4. Nilai Statistik Pengujian
r uji
=
r uji
=
6 d
  1
n n2  1
2


L5. Membuat Keputusan, Tafsiran, dan
Kesimpulan
 Jika nilai statistik pengujian masuk dalam
kawasan tak kritikal maka Ho adalah benar,
maka terima Ho.
L5. Membuat Keputusan, Kesimpulan dan
Tafsiran
 Jika nilai statistik pengujian masuk dalam
kawasan kritikal maka Ho adalah tak benar, maka
Ho ditolak dan seterusnya, Hp diterima (bermakna
ada bukti Hp adalah benar)
Correlation Coefficient - A measure of the
strength and direction of a linear relationship
between two variables
The range of r is from -1 to 1.
-1
If r is close to
-1 there is a
strong
negative
correlation
0
If r is close to
0 there is no
linear
correlation
1
If r is close
to 1 there is
a strong
positive
correlation
Purpose – Determine relationship between two metric variables
Requirement : DV - Interval / Ratio scale and IV- Interval / ratio scale
Descriptive : Magnitude and Direction of Correlation, r
Inferential:
Hypothesis
testing
Hypotheses:
HO: ρp= 0
HA: ρp≠ 0
HO: ρp= 0
ρp> 0
HO: ρp= 0
ρp< 0
 Objektif kajian
 Mengkaji sama ada terdapat perkaitan bilangan
ponteng kelas dengan prestasi dalam ujian.
 Soalan kajian
 Apakah perkaitan antara bilangan ponteng kelas
dengan prestasi dalam ujian
 Hipotesis kajian
 Ho : Tiada terdapat hubungan antara bilangan ponteng
kelas dengan prestasi dalam ujian
 HA : Terdapat hubungan antara bilangan ponteng kelas
dengan prestasi dalam ujian
Application 1
Absences Grade
x
8
2
5
12
15
9
6
y
78
92
90
58
43
74
81
Absences
x
8
2
5
12
15
9
6
Final
Grade
Grade
95
90
85
80
75
70
65
60
55
50
45
40
0
2
4
6
8
10
12
Absences
y
78
92
90
58
43
74
81
14
16
x
Pearson Correlation
rp =
n [xy] - [xy]
[ n  x2 - ( x) 2 ] [ n  y2 - ( y) 2 ]
n = no of pairs
 x y = summation of x multiply by y
 x = summation of x
 y = summation of y
Computation of r
1
2
3
4
5
6
7
x
y
xy
x2
y2
8
2
5
12
15
9
6
78
92
90
58
43
74
81
624
184
450
696
645
666
486
64
4
25
144
225
81
36
6084
8464
8100
3364
1849
5476
6561
3751
579
39898
57 516
r
 3155
804 13030
= - 0.975
Inferential Analysis – Hypothesis Testing
Steps in Hypothesis Testing
1. State the null and alternative hypothesis
HO: ρp= 0
HA:
ρp≠ 0 or ρp> 0 or ρp< 0
2. Determine critical value
df = n – 2 One-tailed or Two-tailed
3. Calculate the test statistic
4. Make your decision
5. Make conclusion
Reject H : Significant relationship between the two
variables
Fail to Reject Ho: No significant relationship between
the two variables
O
Example 1 :
Data were collected from a randomly selected sample to
determine relationship between average assignment scores
and test scores in statistics. Distribution for the data is
presented in the table below. Assuming the data are normally
distributed.
1. Calculated an appropriate correlation
Data set:
coefficient.
2. Describe the nature of relationship
between the two variable.
3. Test the hypothesis on the relationship
at 0.01 level of significance.
Assign
8.5
6
9
10
8
7
5
6
7.5
5
Test
88
66
94
98
87
72
45
63
85
77
Pearson Correlation
rp =
10 [ 5795.5 ] - [ 72 x775 ]
[ 10 (544.5) - (72) 2 ] [ 10 (62441)- (775) 2 ]
r = 0.8649
Inferential Analysis – Hypothesis Testing
Steps in Hypothesis Testing
1. State the null and alternative hypothesis
HO: ρ p = 0,
HA: ρ p ≠ 0
2. Calculate the test statistic:
3. Determine critical value: df = n – 2, Two-tailed .
4. Make your decision:
5. Make conclusion:
Calculate the test statistic
X
Y
XY
X2
Y2
8.5
88
748
72.25
7744
6
66
396
36
4356
9
94
846
81
8836
10
98
980
100
9604
8
87
696
64
7569
7
72
504
49
5184
5
45
225
25
2025
6
63
378
36
3969
7.5
85
637.5
56.25
7225
5
77
385
25
5929
Steps in Hypothesis Testing
1. State the null and alternative hypothesis
HO: ρ p = 0, HA: ρ p ≠ 0
2. Calculate the test statistics: r = .865
3. Determine critical value: df = n – 2, Two-tailed.
r critical= 0.765
4. Make your decision: r cal > r critical so reject null
hypothesis, accept alternative hypothesis
5. Make conclusion: There is significant relationship
between assignment scores and test scores r (8) =
0.87, p<0.01
Example 2 :
A clinical psychologist hypothesizes a correlation between
a personality dimension (Extroversion/introversion) and
depression. Two questionnaires are administered. One
measures the personality trait, with bigger scores indicating
more extroversion and lower scores tending toward
introversion. The other measures depression, with bigger
scores reflecting greater depression. Lower scores on the
depression inventory are not indicative of depression.
1. Calculated and describe the appropriate correlation
coefficient.
3. Test the hypothesis on the relationship at 0.05 level of
significance.
Calculate the test statistic
X
Y
X2
XY
Y2
8
6
9
6.5
4
3.5
52
24
31.5
64
36
81
42.25
16
12.25
10
2
4
3.5
6
7.5
35
12
30
100
4
16
12.25
36
56.25
5
6
7
4.5
2
2.5
22.5
12
17.5
25
36
49
20.25
4
6.25
3
60
5
45
15
251.5
9
420
25
230.5
Steps in Hypothesis Testing
1. State the null and alternative hypothesis
HO: ρ p = 0, HA: ρ p ≠ 0
2. Calculate the test statistic:
r = -0.451
3. Determine critical value: df = n – 2, Two-tailed. r critical = 0.67
4. Make your decision: r cal < r critical so reject null hypothesis
accept alternative hypothesis.
5. Make conclusion: There is no significant relationship between
extroversion and depression measurement, r (8) = - .45 ,
p>0.05. Therefore we cannot conclude that those people who
are extroverts are more likely to be depressed, whilst those
people who are introverts are more likely not depressed.
Spearman’s rank correlation coefficient
 Non parametric method:
 Less power but more robust.
 Does not assume normal distribution.
 The correlation coefficient also varies between -1 and 1
Purpose – Determine relationship between two rank scores
Requirement : variables measured at ordinal scale and ordinal scale,
ordinal scale and interval scale, ordinal scale and ratio scale
Descriptive
Direction
Correlation Coefficient,
Inferential
r
Hypotheses:
HO: ρs = 0
HA: ρs ≠ 0
ρs > 0
ρs < 0
Strength / Magnitude
Spearman rho Correlation
r =
1
-
[6D2]
n [ n2 - 1 ]
6 d
  1
2
n n 1
2

n = no of pairs
 D = summation of the differences between pairs of scores

As a measure of popularity, the classroom teacher is asked to rank
all of the children from the most to least popular. To measure
dominance, the children are given the opportunity to play a new video
game, but they must come to an agreement about which of them will go
first, which will go second, and so on. The order which the children
play the game is taken as the dominance hierarchy. A rank of 1 on
Dominance is assigned to the most dominant child and A rank of 1is
assigned to the most on popular child.
1. Calculate the appropriate correlation coefficient
2. Describe the nature of relationship between the two variables
3. Test the hypothesis on the relationship at 0.05 level of significance
Example for Spearman rho
Dominance
X
1
Popularity
Y
1
7
6
8
3
4
8
9
5
3
3
5
10
4
6
9
2
7
10
D
D2
0
-1
-3
3
0
1
1
4
2
-8
0
1
9
9
0
1
1
16
4
64
r
=
1 -
[6D2]
n [ n2 - 1 ]
r
=
r
1 -
[ 6(105 )]
630
10 [ 100 - 1 ]
990
= 1 – 0.642
r = 0.36
2. There is a positive and weak relationship between
dominance and popularity.
3. Test the hypothesis on the relationship between the two variable
at 0.05 level of significance.
a. State the null and alternative hypotheses
H O : ρs = 0
H A : ρs ≠ 0
b. rs = 0.36
c. Determine critical value
Critical rs = 0.648
d. Decision: Since calculated rs (0.36) is smaller than critical
rs (0.648.), fail to reject the null hypothesis, accept null
hypothesis.
e. Conclusion
Conclude there is no significant relationship between dominance
and popularity at 0.05 level of significance, rs =0.36, p> .05.
Results showed that the more popular the child does not mean he
or she is more likely to assume a dominant position in the class.
Example2:
Data solicited from a randomly
selected sample were used to
measure relationship
between working environment
and work commitment. EDA
revealed the work commitment
data did not meet the assumption
of normality.
1. Calculate and describe the
appropriate correlation coefficient
3. Test the hypothesis on the
relationship at 0.05 level of
significance
ID
1
2
3
4
5
6
7
8
9
10
11
12
X
6
5
8
4
9
6
7
5
4
7
6
7
Y
15
34
11
9
17
12
13
23
11
12
14
16
r
=
1 -
[ 6 (242.5)]
12 [ 144 - 1 ]
1455
1716
r = 0.1521
r2 = COEFFICIENT OF DETERMINATION – TELL
YOU HOW MUCH VARIATION OF Y IS EXPLAINED
BY X
X –WORK
ENV
Y-WORK
COMM
6
15
5
34
8
11
4
9
9
17
6
12
7
13
5
23
4
11
7
12
6
14
7
16
Pangkat
untuk X
Pangkat
untuk Y
6
D
D2
8
-2
4
3.5
12
-8.5
72.25
11
2.5
8.5
72.25
1.5
1
0.5
0.25
12
10
2
4
6
4.5
1.5
2.25
9
6
3
9
3.5
11
-7.5
56.25
1.5
2.5
-1.0
1
9
4.5
4.5
20.25
6
7
-1.0
1
9
9
0
0
242.5
3. Test the hypothesis on the relationship between the two variable
at 0.05 level of significance.
a. State the null and alternative hypotheses
H O : ρs = 0
H A : ρs ≠ 0
b. rs = 0. 587
c. Determine critical value
Critical rs = 0.887
d. Decision: Since calculated rs (0.887 is larger than critical
rs (0.587.), we reject the null hypothesis, accept alternative
hypothesis.
e. Conclusion
Conclude there ino significant relationship between perception
towards work environment with level of work commitment at 0.05
level of significance, rs =0.36, p< .05. Results showed that the
positive and high perception on work environment has positive
impact on work commitment of employees.
Uji diri anda!!!-Apakah pengujian statistik yang
diperlukan dan seterusnya jalankan analisis
yang diperlukan
EXAMPLE DATA
Parents Marital Children Marital Performance
Satisfaction
Satisfaction
Subject
1
1
3
70
2
3
2
80
3
7
6
40
4
9
7
35
5
8
8
50
6
4
6
40
7
5
3
30
Subjek
Pangkat
Agresif
Pangkat
Agresif
1
8
14
2
10
12
3
4
9
4
1
4
5
5
11
6
6
10
7
3
1
8
9
12
9
7
10
10
2
4
CONTOH DATA 3
Jantina
Tahap
Stail
Kepemimpinan Kepimpinan
Persepsi
Prestasi oleh
Guru
1
18
Autokratik
20
1
20
Autokratik
30
1
24
Autokratik
40
1
11
Demokratik
85
1
15
Demokratik
70
2
16
Demokratik
30
2
12
Demokratik
80
2
19
Autokratik
40
2
17
Demokratik
25
2
22
Autokratik
75
Point-biserial Correlation
Purpose – Determine relationship between A DICHOTOMOUS
VARIABLE (2 categories) and a continuous variable (interval/ratio
data)
Requirement : normally distributed continuous variables and
independent variable with only two categories (ex: male/female,
high/low, yes/no, part-time/full-time)
r =
•
•
•
•
•
•
y1 – y2
[ n1 n2 ]
sy
n[n-1]
Mean of group 1
Mean of group 2
Std dev of continuous variable
No of subjects in group 1
No of subjects in group 2
Total no of subjects
Example1:
A psychologist hypothesizes
an association between
marital status and need for
achievement. A questionnaire
measuring need for
achievement is administered
to married and single people.
1. Calculate the appropriate
correlation coefficient
2. Describe the nature of
relationship between the
two variables.
3. Test the hypothesis on the
relationship at 0.05 level
of significance
Marital status
2
2
1
1
1
2
1
2
2
1
1
2
1
1
Need for Achievement
3
7
12
16
24
11
15
10
11
18
22
9
19
17
Point-biserial Correlation
r =
•
•
•
•
•
•
y1 – y2
[ n1 n2 ]
sy
n[n-1]
Mean of married subject = 8.5 (group 2)
Mean of single subjects = 17.9 (group 1)
Std dev of need of achievement scores = 5.89
No of married subjects = 6
No of single subjects = 8
Total no of subjects = 14
Calculate
the
test
statistic
y
y
2
3
9
7
49
12
144
16
256
24
576
11
121
15
225
10
100
11
121
18
324
22
484
9
81
19
361
17
289
194
3140
S =  y2 - (y) 2/ n
n-1
S =
3140 - 1942 / 14
13
S = 5.8947
Point-biserial Correlation
r =
17.9 – 8.5
5.89
r pb = (1.595) (0.514)
r pb = 0.82
[8x6]
14 [ 14 - 1 ]
The mean need for achievement for single
individual is 17.9 and for married
individuals is 8.5. There is a strong
relationship between marital status and
need for achievement with correlation
coefficient of 0.82. Need of achievement for
single individual is higher than married
individual.
3. Test the hypothesis on the relationship between the
two variable at 0.05 level of significance.
a.
State the null and alternative hypotheses
HO : ρ pb = 0
HA : ρ pb ≠ 0
b.
r pb = 0.82
c.
Determine critical value
Critical r pb = 0.532
d.
Decision
Since calculated r pb (0.82) is greater than critical r pb (0.532),
reject the null hypothesis thus accept alternative hypothesis.
e.
Conclusion
Conclude there is significant relationship between
marital status and need for achievement, r pb (12)=.82, p<0.05
Findings indicated that single individuals showed a higher
need for achievement compared to married individuals.