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Pertemuan 2 1 dan 22 Analisis Regresi dan Korelasi Sederhana

Pertemuan 2 1 dan 22 Analisis Regresi dan Korelasi Sederhana. Matakuliah : I0284 - Statistika Tahun : 200 8 Versi : Revisi. Learning Outcomes. Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu :

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Pertemuan 2 1 dan 22 Analisis Regresi dan Korelasi Sederhana

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  1. Pertemuan 21 dan 22Analisis Regresi dan Korelasi Sederhana Matakuliah : I0284 - Statistika Tahun : 2008 Versi : Revisi

  2. Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa akan dapat menghitung dugaan parameter regresi sederhana, korelasi dan menguji keberartiannya.

  3. Outline Materi • Estimasi koefisien regresi • Inferensia parameter regresi • Koefisien korelasi • Koefisien determinasi • Inferesia koefisien korelasi

  4. Persamaan Regresi • Persamaan matematika yang memungkinkan kita meramalkan nilai-nilai peubah tak bebas dari nilai-nilai satu atau lebih peubah bebas disebut Persamaan Regresi • Persamaan Regresi Sederhana:

  5. Testing for Significance • To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of b1 is zero. • Two tests are commonly used • t Test • F Test • Both tests require an estimate of s2, the variance of e in the regression model.

  6. Testing for Significance • An Estimate of s2 The mean square error (MSE) provides the estimate of s2, and the notation s2 is also used. s2 = MSE = SSE/(n-2) where:

  7. Testing for Significance • An Estimate of s • To estimate s we take the square root of s 2. • The resulting s is called the standard error of the estimate.

  8. Testing for Significance: t Test • Hypotheses H0: 1 = 0 Ha: 1 = 0 • Test Statistic • Rejection Rule Reject H0 if t < -tor t > t where tis based on a t distribution with n - 2 degrees of freedom.

  9. Contoh Soal: Reed Auto Sales • t Test • Hypotheses H0: 1 = 0 Ha: 1 = 0 • Rejection Rule For  = .05 and d.f. = 3, t.025 = 3.182 Reject H0 if t > 3.182 • Test Statistics t = 5/1.08 = 4.63 • Conclusions Reject H0

  10. Confidence Interval for 1 • We can use a 95% confidence interval for 1 to test the hypotheses just used in the t test. • H0 is rejected if the hypothesized value of 1 is not included in the confidence interval for 1.

  11. Confidence Interval for 1 • The form of a confidence interval for 1 is: where b1 is the point estimate is the margin of error is the t value providing an area of a/2 in the upper tail of a t distribution with n - 2 degrees of freedom

  12. Contoh Soal: Reed Auto Sales • Rejection Rule Reject H0 if 0 is not included in the confidence interval for 1. • 95% Confidence Interval for 1 = 5 +- 3.182(1.08) = 5 +- 3.44 / or 1.56 to 8.44/ • Conclusion Reject H0

  13. Testing for Significance: F Test • Hypotheses H0: 1 = 0 Ha: 1 = 0 • Test Statistic F = MSR/MSE • Rejection Rule Reject H0 if F > F where F is based on an F distribution with 1 d.f. in the numerator and n - 2 d.f. in the denominator.

  14. Example: Reed Auto Sales • F Test • Hypotheses H0: 1 = 0 Ha: 1 = 0 • Rejection Rule • For  = .05 and d.f. = 1, 3: F.05 = 10.13 • Reject H0 if F > 10.13. • Test Statistic • F = MSR/MSE = 100/4.667 = 21.43 • Conclusion • We can reject H0.

  15. Some Cautions about theInterpretation of Significance Tests • Rejecting H0: b1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a cause-and-effect relationship is present between x and y. • Just because we are able to reject H0: b1 = 0 and demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y.

  16. Using the Estimated Regression Equationfor Estimation and Prediction • Confidence Interval Estimate of E(yp) • Prediction Interval Estimate of yp yp+t/2 sind where the confidence coefficient is 1 -  and t/2 is based on a t distribution with n - 2 d.f.

  17. Contoh Soal: Reed Auto Sales • Point Estimation If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: y = 10 + 5(3) = 25 cars • Confidence Interval for E(yp) 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is: 25 + 4.61 = 20.39 to 29.61 cars • Prediction Interval for yp 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: 25 + 8.28 = 16.72 to 33.28 cars ^

  18. ^ Residual Analysis • Residual for Observation i yi – yi • Standardized Residual for Observation i where: ^ ^ ^

  19. Contoh Soal: Reed Auto Sales • Residuals

  20. Contoh Soal: Reed Auto Sales • Residual Plot

  21. Korelasi Linear • Koefisien korelasi linear didefiisikan sebagai ukuran hubungan linear antara dua peubah X dan Y, dan dilambangkan dengan r. • Ukuran hubungan linear antara dua peubah X dan Y diduga dengan koefisien korelasi contoh r yaitu • Koefisien determinasi = r2

  22. Uji Korelasi Sederhana • Hipotesis: • Ho : r = 0 (tidak ada hubungan x dan y) • Ha : r > 0, r < 0, atau r  0 • Statistik uji:

  23. Selamat Belajar Semoga Sukses.

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