STATISTIK PENDIDIKAN EDU5950 SEM1 2013-14

1. STATISTIK PENDIDIKANEDU5950SEM1 2013-14 STATISTIK INFERENSI:PENGUJIAN HIPOTESIS BAGI ANALISIS KORELASI DAN REGRESI (UJIAN – rP , rS, rPb ) Rohani Ahmad Tarmizi - EDU5950

2. Analisiskorelasidigunakanuntukmenjawabpersoalankajiansepertiberikut: • Adakahterdapathubunganantaraduapembolehubahtersebut? • “Is there relationship between the two variables?” • Sejauhmanakahhubungantersebut? • “How strong is the relationship?” • Apakaharahhubungantersebut? • “What is the direction of the relationship?”

3. 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”.

4. 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.

5. Dua Cara Menentukan Korelasi • 1. Secara bergambariaitu 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.

6. Dua Cara Menentukan Korelasi • 2. Secara berangkaiaitu dengan menentukan pekali, koefisi atau indeks. • Daripada pekali tersebut kita dapat mengetahui keteguhan (magnitud) korelasi tersebut serta arahnya sama positif atau negatif.

7. Scatter Plots and Types of Correlation x = SAT score y = GPA GPA Positive Correlation as x increases y increases

8. Scatter Plots and Types of Correlation Accidents x = hours of training y = number of accidents Negative Correlation as x increases, y decreases

9. Scatter Plots and Types of Correlation x = height y = IQ IQ No linear correlation

10. Analisis Korelasi Menunjukkan 3 perkara penting, iaitu: • Arah/Direction (positive or negative) • Bentuk/Form (linear or non-linear) • Kekuatan/Magnitude (size of coefficient)

11. 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.

12. Pekali Pearson r = n [  x y ] - [  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

13. Pekali Spearman r = 1 - [ 6  B 2 ] n [ n2 - 1 ] n = bilanganpasanganskor  B = jumlahbezaantarasetiappasanganpangkatan

14. Pekali Point-biserial r = y1 – y2[ n1 n2 ] sy n [ n - 1 ]

15. 1 -1 0 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 negative correlation If r is close to 0 there is no linear correlation If r is close to 1 there is a strong positive correlation

16. 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

17. Other Strengths of Association- By Johnson and Nelson (1986) The same strength interpretations hold for negative values of r, only the direction interpretations of the association would change.

18. 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.

19. 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

20. 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

21. 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.

22. L2. TaburanPersampelan • 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

23. 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.

24. 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.

25. L4. NilaiStatistikPengujian r diuji = r diuji =

26. Jika nilai statistik pengujian masuk dalam kawasan tak kritikal maka Ho adalah benar, maka terima Ho. L5. MembuatKeputusan, Kesimpulandantafsiran

27. 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) L5. MembuatKeputusan, KesimpulandanTafsiran

28. 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

29. Calculate the test statistic

30. 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= 0.7646 • Make your decision: r cal > r critical so reject null hypothesis, accept alternative hypothesis • Make conclusion: There is significant relationship between assignment scores and test scores r (8) = 0.87, p<0.01

31. 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

32. 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

33. . 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.

34. 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

36. 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).

39. r = 1 -[ 6  D 2 ] n [ n2 - 1 ] r = 1 -[ 6(50.5 )] 11 [ 121 - 1 ] 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.

40. 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 = 0.618 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.

41. Point-biserial Correlation rpb = 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

42. 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. • Calculate the appropriate • correlation coefficient • Describe the nature of • relationship between the two variables. • 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

43. Point-biserial Correlation 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

44. Point-biserial Correlation r = 17.9 – 8.5 [ 8 x 6 ] 5.89 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. r pb = 0.82

45. 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.

46. ANALISIS REGRESI Analisis 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.

47. 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.

48. 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

49. KECERUNAN GARIS REGRESI b = n [  x y ] - [  x  y ] [ n  x2 - ( x)2 ] n = bilanganpasanganskor  x y = jumlahskor x didarabdenganskor y  X = jumlahskor x  y = jumlahskor y

50. a = PINTASAN PADA PAKSI Y a = y – b x