STATISTIK INFERENSI:PENGUJIAN HIPOTESIS BAGI

ANALISIS KORELASI DAN REGRESI

(UJIAN – rP , rS , rPb )Rohani Ahmad Tarmizi - EDU5950 1

Analisis korelasi 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 KORELASIAnalisis 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 bekerjaAmalan 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.

Positive Correlationas x increases y increases

x = SAT scorey = GPAGPA

Scatter Plots and Types of Correlation

Accidents

Negative Correlationas x increases, y decreases

x = hours of trainingy = number of accidents

Scatter Plots and Types of Correlation

IQ

No linear correlation

x = heighty = IQ

Scatter Plots and Types of Correlation

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 correlationDigunakan apabila p/u x dan y adalah pada skala

ordinal atau gabungan ordinal dengan sela/nisbah.

Point-biserial correlationDigunakan apabila p/u x adalah dikotomus dan

p/u y adalah pada skala sela atau nisbah.

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

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

n = bilangan pasangan skorn = bilangan pasangan skor

x y = jumlah skor x didarab dengan skor yjumlah skor x didarab dengan skor y

x = jumlah skor xjumlah skor x

y = jumlah skor yjumlah skor y

r = r = 1 1 - - [ 6 B 2 ]

n [ n2 - 1 ]

n = bilangan pasangan skorn = bilangan pasangan skor

B = jumlah beza antara setiap pasangan pangkatanjumlah beza antara setiap pasangan pangkatan

r = r = y y11 – y – y22 [ n11 n22 ]

sy n [ n - 1 ]

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 POSITIF yang

signifikan antara tahap kepimpinan pengajaran Pengetua dengan prestasi akademik sekolah di Sabah

Hipotesis nol/sifar – Tiada terdapat hubungan POSITIF yang

signifikan antara tahap kepimpinan pengajaran Pengetua dengan prestasi akademik sekolah di Sabah

L1. Nyatakan hipotesis

Hipotesis penyelidikan – Terdapat hubungan NEGATIF yang

signifikan antara tahap kepimpinan pengajaran Pengetua dengan BILANGAN MASALAH DISIPLIN sekolah di Sabah

Hipotesis nol/sifar – Tiada terdapat hubungan NEGATIF yang

signifikan antara tahap kepimpinan pengajaran Pengetua dengan BILANGAN MASALAH DISIPLIN 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 PersampelanTaburan 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 KritikalNilai 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 PengujianIni 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.

r diuji =

r diuji =

1

61

2

2

nn

d

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

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 correlationData 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 Test8.5 886 669 9410 988 877 725 456 637.5 855 77

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

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 coefficientNon 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 Y1 1 12 2 13 3 24 4 35 5 46 1 37 2 38 3 29 4 510 5 511 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 = ±.618

Participant Ratings of work

environment

Ratings of work

commitment

Rank of work

environment

RankWork

commitment

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 hypothesis hence accept research hypothesis. Conclusion: There was a statistically significant positive correlation between between ratings of working environment and one’s work commitment among employees (rho = 0.77, p < 0.05, N = 11).

Participant Ratings of work

environment

Ratings of work

commitment

Rank of work

environment

RankWork

commitment

D D2

1 1 1 1.5 0 0

2 2 1 3.5 2 4

3 3 2 5.5 2 4

4 4 3 7.5 1.5 2.25

5 5 4 9.5 1.5 2.25

6 1 3 1.5 -4.5 20.25

7 2 3 3.5 -2.5 6.25

8 3 2 5.5 2 4

9 4 5 7.5 -2.5 6.25

10 5 5 9.5 -.5 0.25

11 6 5 11 1 1

50.5

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 r = = 1 1 -- [ 6 D 2 ]

n [ n2 - 1 ]

rr = = 1 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 = 0HA : ρs ≠ 0

b. rs = 0. 77 c. Determine critical value

Critical rs = 0.618 d. Decision: Since calculated rs (0.77) is larger than critical rs (0.618), 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.

rrpbpb = = yy11 – y – y22 [ n11 n22 ]

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

Example on Point-biserial correlationA 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 = r = y y11 – y – y22 [ n11 n22 ]

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 = r = 17.9 – 8.5 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 = 0HA : 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. ConclusionTherefore 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.

ProsedurnyaANALISIS REGRESI MUDAH terdiri

daripada:Melakarkan gambarajah sebaran bagi

taburan pasangan skor tersebut Menentukan persamaan bagi garis

regresi tersebutPersamaan ini juga dipanggil model

regresiPersamaan/model bagi garis ini ialah

Y’ = a + bxDan 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 + bxY’ = Nilai anggaran bagi yb = kecerunan bagi garis

tersebuta = pintasan pada paksi y

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

[ n x2 - ( x)2 ]

n = bilangan pasangan skorn = bilangan pasangan skor

x y = jumlah skor x didarab dengan skor yjumlah skor x didarab dengan skor y

X = jumlah skor xjumlah skor x

y = jumlah skor yjumlah skor y

a = y – b x

X Y

12 8

2 3

1 4

6 6

5 9

8 6

4 6

15 22

11 14

13 6

   

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

         

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 + aY’ = Nilai anggran bagi yb = kecerunan bagi garis

tersebuta= pintasan pada paksi y

r= 0.70. Ini menunjukkan bahawa 49% variasi

dalam y adalah sumbangan daripada XKecerunannya 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 + 2.1Jika x=7, maka Y’= 7.84Jika x=10, maka Y’= 10.3Jika x=14, maka Y’=13.58

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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)

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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

ght

(kgs

)

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

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0 50 100 150 200 250

Drug A (dose in mg)

Sym

pto

m I

nd

ex

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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)

Sym

pto

m I

nd

ex

Example: Symptom Index vs Drug A

Graph Four: Relationship between Symptom Index and Drug B (with best-fit line)

0

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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 lineThe slope of the regression lineThe 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 linea 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 populationThe cases represents a random sample from the population

Simple RegressionHow best to summarise the data?

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0 50 100 150 200 250

Drug A (dose in mg)S

ym

pto

m I

nd

ex

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0 50 100 150 200 250

Drug A (dose in mg)

Sym

pto

m I

nd

ex

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

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120

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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 Regression R2 - “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 data

0

50

100

150

200

250

300

0 100 200 300

IV (regressor, predictor)

DV

Simple Regression Low 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

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150

200

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0 100 200 300

IV

DV

0

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0 50 100 150 200 250

IV

DV

Simple Regression High values of R2

0

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0 50 100 150 200 250

Drug A (dose in mg)

Sym

ptom

In

dex

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120

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0 50 100 150 200 250

Drug B (dose in mg)

Sym

ptom

In

dex

Good fit R2 high

High variance explained

Moderate fit R2 lower

Less variance explained

Simple Regression R2 - “Goodness of fit”

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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

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“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)

Sym

pto

m I

nd

ex

Example: Symptom Index vs Drug A

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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 Test1 8.5 882 6 663 9 944 10 985 8 876 7 727 5 458 6 639 7.5 8510 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 Y1 8.5 882 6 663 9 944 10 985 8 876 7 727 5 458 6 639 7.5 8510 5 77

Summary stat:

n 10ΣΧ 72ΣΥ 775ΣΧ² 544.5ΣΥ² 62,441ΣΧΥ 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

ΔY8.257

18.05

MARITAL SATISFACTION

Parents : X Children : Y

1 33 27 69 78 84 65 3

Mean of X Mean of YNo 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

ΔY0.65

8.43

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 rendahData 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

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