statistik inferensi: pengujian hipotesis bagi...

STATISTIK INFERENSI:PENGUJIAN HIPOTESIS BAGI ANALISIS KORELASI

DAN REGRESI

(UJIAN – rP , rS , rPb )

Rohani Ahmad Tarmizi - EDU5950 1

Analisis korelasi digunakan untuk menjawabpersoalan kajian seperti berikut:

Adakah terdapat hubungan antaradua 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 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:

Keyakinan dalam mentadbir dengan prestasi kepimpinan dalam kalangan pengetua

Persepsi guru kanan dan staff pentadbiran terhadap tahap kepimpinan pengetua di sekolah

Umur dengan kepuasan bekerja

Amalan pemakanan pangkat keyakinan untuk menyertai marathon.

Dua Cara Menentukan Korelasi

1. 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

2. 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.

300 350 400 450 500 550 600 650 700 750 800

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

4.00

Math SAT

Positive Correlationas x increases y increases

x = SAT score

y = GPAGPA

Scatter Plots and Types of Correlation

0 2 4 6 8 10 12 14 16 18 20

0

10

20

30

40

50

60

Hours of Training

Accid

ents

Accidents

Negative Correlationas x increases, y decreases

x = hours of training

y = number of accidents


807672686460

160

150

140

130

120

110

100

90

80

Height

IQ

IQ

No linear correlation

x = height

y = IQ


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.

r = n [ x y ] - [ x y ]

[ n x2 - ( x) 2 ] [ n y2 - ( y) 2 ]

Pekali Pearson

n = bilangan pasangan skor

x y = jumlah skor x didarab dengan skor y

x = jumlah skor x

y = jumlah skor y

r = 1 - [ 6 B 2 ]

n [ n2 - 1 ]

Pekali Spearman


B = jumlah beza antara setiap pasangan pangkatan

r = y1 – y2 [ n1 n2 ]

sy n [ n - 1 ]

Pekali Point-biserial

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.

If r is close

to 1 there is

a strong

positive

correlation

If r is close to

-1 there is a

strong

negative

correlation

If r is close to

0 there is no

linear

correlation

-1 0 1

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 Association-By 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

The same strength interpretations hold for negative values of r, only the direction

interpretations of the association would change.

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.

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 diuji =

r diuji =

16

12

2

nn

d

L5. Membuat Keputusan, Kesimpulan dantafsiran

Jika nilai statistik pengujian masuk dalam kawasan tak kritikal maka Ho adalah benar, maka terima Ho.

L5. Membuat Keputusan, Kesimpulan danTafsiran

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)

Example of Pearson correlation

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

coefficient.

2. Describe the nature of relationship

between the two variable.

3. Test the hypothesis on the relationship

at 0.01 level of significance.

Data set:

Assign Test

8.5 88

6 66

9 94

10 98

8 87

7 72

5 45

6 63

7.5 85

5 77

Calculate the test statisticX 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

3. Determine critical value: df = n – 2, Two-tailed.

r critical= 0.7646

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

1. State the null and alternative hypothesis

HO: ρ p = 0, HA: ρ p ≠ 0

2. Calculate the test statistics: r = .865

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

Example of Spearman correlation

Data solicited from a randomly

selected sample of employees

were used to measure

relationship between ratings of

working environment and one’s

work commitment.

1. Calculate and describe the

appropriate correlation coefficient

2. Test the hypothesis on the

relationship at 0.05 level of

significance

ID X Y

1 1 1

2 2 1

3 3 2

4 4 3

5 5 4

6 1 3

7 2 3

8 3 2

9 4 5

10 5 5

11 6 5

Null hypothesis: There is no significant correlation between

between ratings of working environment and one’s work

commitment among work employees.

Research hypothesis: There is significant correlation

between between ratings of working environment and

one’s work commitment among work employees.

.

Null hypothesis is true

Research hypothesis is true Research hypothesis is true

Determined the critical values in the sampling distribution. Degrees of freedom

From Table r, r = ±.456

Participant Ratings of

work

environment

Ratings of

work

commitment

Rank of

years

Rank

of rating

D D2

1 1 1 1.5 1.5 0 0

2 2 1 3.5 1.5 2 4

3 3 2 5.5 3.5 2 4

4 4 3 7.5 6 1.5 2.25

5 5 4 9.5 8 1.5 2.25

6 1 3 1.5 6 -4.5 20.25

7 2 3 3.5 6 -2.5 6.25

8 3 2 5.5 3.5 2 4

9 4 5 7.5 10 -2.5 6.25

10 5 5 9.5 10 -.5 0.25

11 6 5 11 10 1 1

50.5

Make a decision: Reject the null hypothesishence accept research hypothesis.Conclusion: There was a statistically significantpositive correlation between between ratings ofworking environment and one’s workcommitment among employees (rho = 0.77, p <0.05, N = 11).

r = 1 – 0.229

r = 0.77

There is a positive and strong relationship between ratings

of working environment and one’s work commitment

among employees.

r = 1 - [ 6 D 2 ]

n [ n2 - 1 ]

r = 1 - [ 6(50.5 )]

11 [ 121 - 1 ]

2. Test the hypothesis on the relationship between the two

variables at 0.05 level of significance.

a. State the null and alternative hypotheses

HO : ρs = 0

HA : ρs ≠ 0

b. rs = 0. 77

c. Determine critical value

Critical rs = 0.456

d. Decision: Since calculated rs (0.77) is larger than critical

rs (0.456), we reject the null hypothesis, accept alternative

hypothesis.

e. Conclusion

Conclude there is significant relationship between ratings towards

work environment with level of work commitment at 0.05 level of

significance, rs (11) = 0.77, p< .05. Results showed that the positive

and high perception on work environment has positive impact on

work commitment among employees.

rpb = y1 – y2 [ n1 n2 ]

sy n [ n - 1 ]

Point-biserial Correlation

• 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

Example on Point-biserial

correlation

A psychologist hypothesizes an

association between marital

status (1-single, 2-married) 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 Need for Achievement

2 3

2 7

1 12

1 16

1 24

2 11

1 15

2 10

2 11

1 18

1 22

2 9

1 19

1 17

r = y1 – y2 [ n1 n2 ]

sy n [ n - 1 ]


• Mean of married subject = 8.5

• Mean of single subjects = 17.9

• Std dev. of need of achievement scores = 5.89

• No of married subjects = 6 (2)

• No of single subjects = 8 (1)

• Total no of subjects = 14

r = 17.9 – 8.5 [ 8 x 6 ]

5.89 14 [ 14 - 1 ]


r pb = 0.82

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.

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 value, r pb (0.532), we can reject the null hypothesis

thus accept alternative hypothesis.

e. Conclusion

Therefore there is a significant relationship between

marital status and need for achievement, r pb (12)=.82,

p<0.05. Findings also indicated that single individuals

showed a higher need for achievement compared to

married individuals. Hence marital status has an influence

on one’s need for achievement.

ANALISIS REGRESIAnalisis regresi adalah lanjutan daripada

analisis korelasi dimana sesuatu hubungan telah diperoleh.

Analisis regresi dilaksanakan setelah suatu pola hubungan linear dijangkakan serta suatu pekali ditentukan bagi menunjukkan terdapat hubungan yang linear antara dua pembolehubah.

Selanjutnya bolehlah kita menelah atau meramal sesuatu pembolehubah (p/u criterion) setelah pembolehubah yang kedua (p/u predictive) diketahui.

Prosedurnya ANALISIS REGRESI MUDAH terdiri daripada:

Melakarkan gambarajah sebaran bagi taburan pasangan skor tersebut

Menentukan persamaan bagi garis regresi tersebut

Persamaan ini juga dipanggil model regresi

Persamaan/model bagi garis ini ialah

Y’ = a + bx Dan selanjutnya dengan mengguna

persamaan tersebut, nilai y boleh ditentukan bagi sesuatu nilai x yang telah ditentukan dan juga disebaliknya.

PERSAMAAN BAGI GARIS REGRESI(LEAST-SQUARES REGRESSION LINE)

Y’ = a + bx

Y’ = Nilai anggaran bagi y

b = kecerunan bagi garis tersebut

a = pintasan pada paksi y

b = n [ x y ] - [ x y ]

[ n x2 - ( x)2 ]

KECERUNAN GARIS REGRESI


x y = jumlah skor x didarab dengan skor y

X = jumlah skor x

y = jumlah skor y

a = PINTASAN PADA PAKSI Y

a = y – b x

Data: Tahap kepemimpinan pengetua dengan persepsi

guru terhadap tahap kepemimpinan pengetua

X Y

12 8

2 3

1 4

6 6

5 9

8 6

4 6

15 22

11 14

13 6

PENGIRAAN ANALISIS REGRESI

X Y XY X2 Y2

12 8

2 3

1 4

6 6

5 9

8 6

4 6

15 22

11 14

13 6

PENGIRAAN ANALISIS REGRESI

X Y XY X2 Y2

12 8 96 144 64

2 3 6 4 9

1 4 4 1 16

6 6 36 36 36

5 9 45 25 81

8 6 48 64 36

4 6 24 16 36

15 22 330 225 484

11 14 154 121 196

13 6 78 169 36

77 84 821 805 994

PERSAMAAN BAGI GARIS REGRESI(LEAST-SQUARES REGRESSION LINE)

Y’ = bx + a

Y’ = Nilai anggran bagi y

b = kecerunan bagi garis tersebut

a= pintasan pada paksi y

r= 0.70. Ini menunjukkan bahawa 49% variasi dalam y

adalah sumbangan daripada X Kecerunannya ialah 0.82Min bagi x ialah 7.7Min bagi y ialah 8.4 a = 2.1 (pintasan di paksi y)Model regresi ialah Y’ = .82x + 2.1 Jika x=7, maka Y’= 7.84 Jika x=10, maka Y’= 10.3 Jika x=14, maka Y’=13.58

54

Regression & CorrelationA correlation measures the “degree of

association” between two variables (interval (50,100,150…) or ordinal (1,2,3...))

Associations can be positive (an increase in one variable is associated with an increase in the other) or negative (an increase in one variable is associated with a decrease in the other)

55

Example: Height vs. WeightGraph One: Relationship between Height

and Weight

0

20

40

60

80

100

120

140

160

180

0 50 100 150 200

Height (cms)

Wei

gh

t (k

gs)

Strong positive correlation

between height and weight

Can see how the

relationship works, but

cannot predict one from the

other

If 120cm tall, then how

heavy?

Example: Symptom Index vs Drug A

Strong negative correlation

Can see how relationship works, but cannot make predictions

What Symptom Index might we predict for a standard dose of 150mg?

Graph Two: Relationship between Symptom

Index and Drug A

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250

Drug A (dose in mg)

Sy

mp

tom

In

dex

57

Correlation examples

Regression analysis procedures have as their primary purpose the development of an equation that can be used for predicting values on some DV for all members of a population.

A secondary purpose is to use regression analysis as a means of explaining causal relationships among variables.

Regression

The most basic application of regression analysis is the bivariate situation, to which is referred as simple linear regression, or just simple regression.

Simple regression involves a single IV and a single DV.

Goal: to obtain a linear equation so that we can predict the value of the DV if we have the value of the IV.

Simple regression capitalizes on the correlation between the DV and IV in order to make specific predictions about the DV.

The correlation tells us how much information about the DV is contained in the IV.

If the correlation is perfect (i.e r = ±1.00), the IV contains everything we need to know about the DV, and we will be able to perfectly predict one from the other.

Regression analysis is the means by which we determine the best-fitting line, called the regression line.

Regression line is the straight line that lies closest to all points in a given scatterplot

This line sometimes pass through the centroid of the scatterplot.

“Best fit line”

Allows us to describe relationship between variables more accurately.

We can now predict specific values of one variable from knowledge of the other

All points are close to the line

Graph Three: Relationship between

Symptom Index and Drug A

(with best-fit line)

0

20

40

60

80

100

120

140

160

180

0 50 100 150 200 250

Drug A (dose in mg)

Sy

mp

tom

In

dex


Graph Four: Relationship between Symptom

Index and Drug B


0

20

40

60

80

100

120

140

160

0 50 100 150 200 250

Drug B (dose in mg)

Sym

pto

m I

nd

ex

We can still predict specific values of one variable from knowledge of the other

Will predictions be as accurate?

Why not?

“Residuals”

Example: Symptom Index vs Drug B

3 important facts about the regression line must be known:The extent to which points are scattered around the line

The slope of the regression line

The point at which the line crosses the Y-axis

The extent to which the points are scattered around the line is typically indicated by the degree of relationship between the IV (X) and DV (Y).

This relationship is measured by a correlation coefficient – the stronger the relationship, the higher the degree of predictability between X and Y.

The degree of slope is determined by the amount of change in Y that accompanies a unit change in X.

It is the slope that largely determines the predicted values of Y from known values for X.

It is important to determine exactly where the regression line crosses the Y-axis (this value is known as the Y-intercept).

The regression line is essentially an equation that express Y as a function of X.

The basic equation for simple regression is:

Y = a + bX

where Y is the predicted value for the DV,

X is the known raw score value on the IV,

b is the slope of the regression line

a is the Y-intercept

Simple Linear Regression

♠ Purpose

To determine relationship between two metric variables

To predict value of the dependent variable (Y) based on

value of independent variable (X)

♠ Requirement :

DV Interval / Ratio

IV Internal / Ratio

♠ Requirement :

The independent and dependent variables are normally

distributed in the population

The cases represents a random sample from the population

Simple RegressionHow best to summarise the data?

0

20

40

60

80

100

120

140

160

180

0 50 100 150 200 250

Drug A (dose in mg)S

ymp

tom

In

dex

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250

Drug A (dose in mg)

Sym

ptom

In

dex

Adding a best-fit line allows us to describe data

simply

Establish equation for the best-fit line:

Y = a + bX

General Linear Model (GLM)How best to summarise the data?

0

20

40

60

80

100

120

140

160

180

200

0 50 100 150 200 250

Where: a = y intercept

(constant)

b = slope of best-fit line

Y = dependent variable

X = independent variable

For simple regression, R2 is the square of the correlation coefficient

Reflects variance accounted for in data by the best-fit line

Takes values between 0 (0%) and 1 (100%)

Frequently expressed as percentage, rather than decimal

High values show good fit, low values show poor fit

Simple RegressionR2 - “Goodness of fit”

R2 = 0

(0% - randomly scattered points, no apparent relationship between X and Y)

Implies that a best-fit line will be a very poor description of data0

50

100

150

200

250

300

0 100 200 300

IV (regressor, predictor)

DV

Simple RegressionLow values of R2

R2 = 1

(100% - points lie directly on the line - perfect relationship between X and Y)

Implies that a best-fit line will be a very good description of data

0

50

100

150

200

250

300

0 100 200 300

IV

DV

0

50

100

150

200

250

0 50 100 150 200 250

IV

DV

Simple RegressionHigh values of R2

0

20

40

60

80

100

120

140

160

180

0 50 100 150 200 250

Drug A (dose in mg)

Sym

pto

m I

nd

ex

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250

Drug B (dose in mg)

Sym

pto

m I

nd

ex

Good fit R2 high

High variance explained

Moderate fit R2

lower

Less variance explained

Simple RegressionR2 - “Goodness of fit”

73

Problem: to draw a straight line through the points that best explains the variance

0

1

2

3

4

5

6

7

8

9

0 2 4 6

Line can then be used

to predict Y from X

74

“Best fit line”

allows us to describe relationship between variables more accurately.

We can now predict specific values of one variable from knowledge of the other

All points are close to the line

Graph Three: Relationship between

Symptom Index and Drug A


0

20

40

60

80

100

120

140

160

180

0 50 100 150 200 250

Drug A (dose in mg)

Sy

mp

tom

In

dex


75

Establish equation for the best-fit line:

Y = a + bX

Best-fit line same as regression line

b is the regression coefficient for x

x is the predictor or regressor variable for y

Regression

Regression - Types

Step –Descriptive Analysis

Derive Regression / Prediction equation

● Calculate a and b

a = y – b X

Ŷ = a + bX

Example on regression analysis

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.

1. Calculate coefficient of determination

and the correlation coefficient

2. Determine the prediction equation.

3. Test hypothesis for the slope at 0.05

level of significance

Data set:

Scores

ID Assign Test

1 8.5 88

2 6 66

3 9 94

4 10 98

5 8 87

6 7 72

7 5 45

8 6 63

9 7.5 85

10 5 77

1. Derive Regression / Prediction equation

215.5

26.1= 8.257=

a= y – b x

= 77.5 – 8.257 (7.2)

= 18.050

ID X Y

1 8.5 88

2 6 66

3 9 94

4 10 98

5 8 87

6 7 72

7 5 45

8 6 63

9 7.5 85

10 5 77

Summary stat:

n 10

ΣΧ 72

ΣΥ 775

ΣΧ² 544.5

ΣΥ² 62,441

ΣΧΥ 5,795.5

Prediction equation:

Ŷ = 18.05 + 8.257X

Interpretation of regression equation

Ŷ = 18.05 + 8.257x

For every 1 unit change in X,

Y will change by 8.257 units

ΔX

ΔY18.05

MARITAL SATISFACTION

Parents : X Children : Y

1 3

3 2

7 6

9 7

8 8

4 6

5 3

Mean of X Mean of Y

No of pairs

X Y

X squared X squared

Standard deviation Standard deviation

XY

Example on regression analysis:

1. Derive Regression / Prediction equation

a= y – b x

= 5.00 +.65 (5.29)

= 8.438

Prediction equation:

Ŷ = 8.44 + 65x

Interpretation of regression equation

Ŷ = 8.43 + .65x

For every 1 unit change in X,

Y will change by .65 units

ΔX

ΔY8.43

ANALISIS “CHI-SQUARE”(KUASA-DUA KHI)

Ini juga merupakan analisis hubungan tetapi lebih dikenali sebagai analisis perkaitan (association)

Analisis ini digunakan pakai bagi menentukan perkaitan antara pasangan pembolehubah yang diukur pada skala nominal atau ordinal ataupun jika salah satunya dipadankan dengan data sela dan nisbah.

Dengan itu pembolehubah seperti Bangsa, Jantina, Suka/tidak suka makanan, Tinggi pencapaian/rendah pencapaian, Kebimbangan tinggi/ kebimbangan sederhana/

kebimbangan rendah

Data frekuensi dicerap dengan membilang kejadian (occurance setiap perkara). Sesuai untuk kajian tinjauan

Daripada frekuensi yang dicerap (observed frequency) analisis “chi-square” memberi kita makluman bahawa ada/tiada perkaitan antara kedua-dua pemboleh ubah.

ANALISIS “CHI-SQUARE” (KUASA-DUA KHI)

KATAKANLAH, penyelidik mengumpul maklumat tentang bangsa bagi responden dan juga kategori amalan pemakanan setiap responden,

ATAU penyelidik tinjau pelajar dibeberapa buah sekolah dari segi jantina dan minta/tidak minat kepada aliran sains

ATAU penyelidik tinjau bapa-bapa dan mengumpul maklumat tahap pendidikan (tinggi/ sederhana/ rendah) dan dikaitkan dengan kategori gaji

Bagi ketiga-tiga contoh tersebut analisis yang sesuai dijalankan adalah analisis tak parametrik (analisis kuasa-dua khi)

dan seterusnya dibina jadual kontingensi atau jadual“crosstabulation”.

Daripada frekuensi yang dicerap (observed frequency) analisis “chi-square” memberi kita makluman bahawa ada/tiada perkaitan antara kedua-dua pemboleh ubah.

ANALISIS “CHI-SQUARE”(KUASA-DUA KHI) Terdapat dua cara/kategori – CHI-SQUARE

TEST OF GOODNESS OF FIT dan TEST OF INDEPENDENCE/DEPENDENCE

TEST GOODNESS OF FIT – menjawab persoalan “adakah terdapat perbezaan kadar bagi sesuatu perkara/kejadian/persetujuan”

TEST OF INDEPENDENCE/ DEPENDENCE –menjawab persoalan “adakah terdapat perkaitan/kebersandaran/ hubungan antara dua perkara

ANALISIS “CHI-SQUARE”(KUASA-DUA KHI)

Dapatan bagi analisis ini lazimnya dalam bentuk jadual frekuensi yang dipanggil jadual kontingensi atau jadual “crosstabulation”.

Daripada frekuensi yang dicerap (observed frequency) analisis “chi-square” ini memberi kita makluman bahawa ada/tiada perkaitan yang signifikan antara kedua-dua pembolehubah yang dikaji

Ataupun ada/tiada perbezaan frekuensi yang signifikan antara kategori-kategori yang dikaji.

•Daripada jadual tersebut kita boleh telitikan atau

kajikan sama ada terdapat hubungan atau perkaitan

antara kedua-dua pemboleh ubah tersebut.

•Selanjutnya analisis pengujian hipotesis perlu

dijalankan ia itu untuk menguji terdapatnya perkaitan

antara kedua-dua pemboleh ubah tersebut dengan

signifikan.

•Pengujian hipotesis ini adalah ujian kuasa dua khi.

•Sekiranya, terdapat perkaitan yang signifikan maka

langkah seterusnya adalah dengan menentukan

darjah atau magnitud hubungan tersebut.

•Bagi analisis ini, data adalah dalam bentuk

kekerapan dan sudah semestinya taburan skor

adalah tidak normal.

•Dengan itu taburan ini dipanggil taburan bebas

(distribution-free).

•Ujian ini juga dipanggil ujian tak parametrik oleh

kerana ia tidak bertabur secara normal.

•Sebagai “rule-of-thumb” penggunaan ujian

parametrik digalakkan oleh kerana oleh kerana

“power” atau kekuatannya, walaubagaimana pun jika

data adalah dalam bentuk nominal serta juga terdapat

taburan data yang tidak normal maka ujian tak

parametrik diterima pakai.

•Ujian-ujian parametrik – sign test, Mann-Whitney U

test, Wilcoxon matched-pairs signed ranks, Kruskal-

Wallis, Chi-square.

Uji diri anda!!!-Apakah pengujian statistik yang diperlukan dan seterusnya jalankan analisis

yang diperlukan

EXAMPLE DATA

Parents Marital

Satisfaction

1

3

7

9

8

4

5

Subject

1

2

3

4

5

6

7

Children Marital

Satisfaction

3

2

6

7

8

6

3

Performance

70

80

40

35

50

40

30

Pangkat

Agresif

8

10

4

1

5

6

3

9

7

2

Subjek

1

2

3

4

5

6

7

8

9

10

Pangkat

Agresif

14

12

9

4

11

10

1

12

10

4

CONTOH DATA 3 Tahap

Kepemimpinan

18

20

24

11

15

16

12

19

17

22

Jantina

1

1

1

1

1

2

2

2

2

2

Stail

Kepimpinan

Autokratik

Autokratik

Autokratik

Demokratik

Demokratik

Demokratik

Demokratik

Autokratik

Demokratik

Autokratik

Persepsi

Prestasi oleh

Guru

20

30

40

85

70

30

80

40

25

75

statistik inferensi: pengujian hipotesis bagi...

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