1 Pertemuan 13 Regresi Linear dan Korelasi Matakuliah: I0262 – Statistik Probabilitas Tahun: 2007 Versi: Revisi

2 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Mahasiswa akan dapat memilih statistik uji untuk koefisien regresi dan korelasi.

3 Outline Materi Pengujian koefisien regresi dengan analisis varians Inferensia tentang koefisien korelasi

4 Testing for Significance To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of  1 is zero. Two tests are commonly used –t Test –F Test Both tests require an estimate of  2, the variance of  in the regression model.

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

6 Testing for Significance An Estimate of  –To estimate  we take the square root of  2. –The resulting s is called the standard error of the estimate.

7 Hypotheses H 0 :  1 = 0 H a :  1 = 0 Test Statistic Rejection Rule Reject H 0 if t t  where t  is based on a t distribution with n - 2 degrees of freedom. Testing for Significance: t Test

8 t Test –Hypotheses H 0 :  1 = 0 H a :  1 = 0 –Rejection Rule For  =.05 and d.f. = 3, t.025 = Reject H 0 if t > –Test Statistics t = 5/1.08 = 4.63 –Conclusions Reject H 0 Contoh Soal: Reed Auto Sales

9 Confidence Interval for  1 We can use a 95% confidence interval for  1 to test the hypotheses just used in the t test. H 0 is rejected if the hypothesized value of  1 is not included in the confidence interval for  1.

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

11 Contoh Soal: Reed Auto Sales Rejection Rule Reject H 0 if 0 is not included in the confidence interval for  1. 95% Confidence Interval for  1 = (1.08) = /or 1.56 to 8.44/ Conclusion Reject H 0

12 Testing for Significance: F Test n Hypotheses H 0 :  1 = 0 H 0 :  1 = 0 H a :  1 = 0 H a :  1 = 0 n Test Statistic F = MSR/MSE n Rejection Rule Reject H 0 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.

13 n F Test Hypotheses H 0 :  1 = 0 Hypotheses H 0 :  1 = 0 H a :  1 = 0 H a :  1 = 0 Rejection Rule Rejection Rule For  =.05 and d.f. = 1, 3: F.05 = For  =.05 and d.f. = 1, 3: F.05 = Reject H 0 if F > Reject H 0 if F > Test Statistic Test Statistic F = MSR/MSE = 100/4.667 = Conclusion Conclusion We can reject H 0. Example: Reed Auto Sales

14 Some Cautions about the Interpretation of Significance Tests Rejecting H 0 :  1 = 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 H 0 :  1 = 0 and demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y.

15 n Confidence Interval Estimate of E ( y p ) n Prediction Interval Estimate of y p y p + t  /2 s ind y p + t  /2 s ind where the confidence coefficient is 1 -  and t  /2 is based on a t distribution with n - 2 d.f. Using the Estimated Regression Equation for Estimation and Prediction

16 Point Estimation If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: y = (3) = 25 cars Confidence Interval for E(y p ) 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is: = to cars Prediction Interval for y p 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: = to cars ^ Contoh Soal: Reed Auto Sales

17 Residual for Observation i y i – y i Standardized Residual for Observation i where: Residual Analysis ^^^^

18 Contoh Soal: Reed Auto Sales Residuals

19 Contoh Soal: Reed Auto Sales Residual Plot

20 Residual Analysis Detecting Outliers – An outlier is an observation that is unusual in comparison with the other data. – Minitab classifies an observation as an outlier if its standardized residual value is +2. – This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier. – This rule’s shortcoming can be circumvented by using studentized deleted residuals. – The |i th studentized deleted residual| will be larger than the |i th standardized residual|.

21 Selamat Belajar Semoga Sukses.