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1 Pertemuan 10 Analisis Ragam (Varians) - 1 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 Analysis of Variance and Experimental Design An Introduction to Analysis of Variance Analysis of Variance: Testing for the Equality of k Population Means Multiple Comparison Procedures An Introduction to Experimental Design Completely Randomized Designs Randomized Block Design

5 Analysis of Variance (ANOVA) can be used to test for the equality of three or more population means using data obtained from observational or experimental studies. We want to use the sample results to test the following hypotheses.  H 0 :  1  =  2  =  3  = ... =  k   H a : Not all population means are equal If H 0 is rejected, we cannot conclude that all population means are different. Rejecting H 0 means that at least two population means have different values. An Introduction to Analysis of Variance

6 Assumptions for Analysis of Variance For each population, the response variable is normally distributed. The variance of the response variable, denoted  2, is the same for all of the populations. The observations must be independent.

7 Analysis of Variance: Testing for the Equality of K Population Means Between-Samples Estimate of Population Variance Within-Samples Estimate of Population Variance Comparing the Variance Estimates: The F Test The ANOVA Table

8 Between-Samples Estimate of Population Variance A between-samples estimate of  2 is called the mean square between (MSB). The numerator of MSB is called the sum of squares between (SSB). The denominator of MSB represents the degrees of freedom associated with SSB. = = _ _

9 Within-Samples Estimate of Population Variance The estimate of  2 based on the variation of the sample observations within each sample is called the mean square within (MSW). The numerator of MSW is called the sum of squares within (SSW). The denominator of MSW represents the degrees of freedom associated with SSW.

10 Comparing the Variance Estimates: The F Test If the null hypothesis is true and the ANOVA assumptions are valid, the sampling distribution of MSB/MSW is an F distribution with MSB d.f. equal to k - 1 and MSW d.f. equal to n T - k. If the means of the k populations are not equal, the value of MSB/MSW will be inflated because MSB overestimates  2. Hence, we will reject H 0 if the resulting value of MSB/MSW appears to be too large to have been selected at random from the appropriate F distribution.

11 Test for the Equality of k Population Means Hypotheses H 0 :  1  =  2  =  3  = ... =  k   H a : Not all population means are equal Test Statistic F = MSB/MSW Rejection Rule Reject H 0 if F > F  where the value of F  is based on an F distribution with k - 1 numerator degrees of freedom and n T - 1 denominator degrees of freedom.

12 The figure below shows the rejection region associated with a level of significance equal to  where F  denotes the critical value. Sampling Distribution of MSTR/MSE Do Not Reject H 0 Reject H 0 MSTR/MSE Critical Value FF FF

13 The ANOVA Table Source of Sum of Degrees of Mean Variation Squares Freedom Squares F TreatmentSSTR k - 1 MSTR MSTR/MSE Error SSE n T - k MSE Total SST n T - 1 SST divided by its degrees of freedom n T - 1 is simply the overall sample variance that would be obtained if we treated the entire n T observations as one data set.

14 Fisher’s LSD Procedure Hypotheses H 0 :  i =  j H a :  i  j Test Statistic Rejection Rule Reject H 0 if t t a/2 where the value of t a/2 is based on a t distribution with n T - k degrees of freedom.

15 Hypotheses H 0 :  i =  j H a :  i  j Test Statistic x i - x j Rejection Rule Reject H 0 if |x i - x j | > LSD where Fisher’s LSD Procedure Based on the Test Statistic x i - x j __ _ _ _ _

16 ANOVA Table for a Completely Randomized Design Source of Sum of Degrees of Mean Variation Squares Freedom Squares F Treatments SSTR k - 1 Error SSE n T - k Total SST n T - 1

17 Selamat Belajar Semoga Sukses.

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