Matakuliah Tahun Versi : A0064 / Statistik Ekonomi : 2005 : 1/1 Pertemuan 18 Pembandingan Dua Populasi-2 1 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Membandingkan pengujian sampel besar untuk perbedaan antara dua proporsi populasi dan pengujian untuk kesamaan dua populasi 2 Outline Materi • Pengujian Sampel Besar untuk Perbedaan antara Dua Proporsi Populasi • Sebaran-F dan Uji untuk Kesamaan Dua Ragam Populasi 3 COMPLETE BUSINESS STATISTICS 8-4 5th edi tion 8-5 A Large-Sample Test for the Difference between Two Population Proportions • Hypothesized difference is zero I: Difference between two population proportions is 0 • p1= p2 » H0: p1 -p2 = 0 » H1: p1 -p20 II: Difference between two population proportions is less than 0 • • p1 p2 » H0: p1 -p2 0 » H1: p1 -p2 > 0 Hypothesized difference is other than zero: III: Difference between two population proportions is less than D • McGraw-Hill/Irwin p1 p2+D » H0:p-p2 D » H1: p1 -p2 > D Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-5 BUSINESS STATISTICS 5th edi tion Comparisons of Two Population Proportions When the Hypothesized Difference Is Zero: Test Statistic When the population proportions are hypothesized to be equal, then a pooled estimator of the proportion ( ) pmay be used in calculating the test statistic. A large-sample test statistic for the difference between two population proportions, when the hypothesized difference is zero: z ( p1 p 2 ) 0 1 1 p(1 p) n1 n2 x1 x1 is the sample proportion in sample 1 and p 1 is the sample n1 n1 proportion in sample 2. The symbol p stands for the combined sample where p1 proportion in both samples, considered as a single sample. That is: pˆ McGraw-Hill/Irwin x x n n 1 1 1 2 Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-6 BUSINESS STATISTICS 5th edi tion Comparisons of Two Population Proportions When the Hypothesized Difference Is Zero: Example 8-8 Carry out a two-tailed test of the equality of banks’ share of the car loan market in 1980 and 1995. Population 1: 1980 n1 = 100 x1 = 53 H 0 : p1 p 2 0 H1: p1 p 2 0 z p 1 = 0.53 Population 2: 1995 n 2 = 100 x 2 = 43 p 2 = 0.43 x1 + x 2 53 43 p 0.48 n1 n 2 100 100 McGraw-Hill/Irwin ( p1 p 2 ) 0 p (1 1 1 p ) n1 n2 0.10 0.004992 0.10 0.53 0.43 1 1 100 100 (.48)(.52) 1.415 0.07065 Critical point: z = 1.645 0.05 H 0 may not be rejected even at a 10% level of significance. Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-7 BUSINESS STATISTICS 5th edi tion Example 8-8: Carrying Out the Test Standard Normal Distribution 0.4 f(z) 0.3 0.2 0.1 0.0 -z0.05=-1.645 Rejection Region 0 z z0.05=1.645 Nonrejection Region Test Statistic=1.415 McGraw-Hill/Irwin Rejection Region Since the value of the test statistic is within the nonrejection region, even at a 10% level of significance, we may conclude that there is no statistically significant difference between banks’ shares of car loans in 1980 and 1995. Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 8-8 5th edi tion Example 8-8: Using the Template P-value = 0.157, so do not reject H0 at the 5% significance level. McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-9 BUSINESS STATISTICS 5th edi tion Comparisons of Two Population Proportions When the Hypothesized Difference Is Not Zero: Example 8-9 Carry out a one-tailed test to determine whether the population proportion of traveler’s check buyers who buy at least $2500 in checks when sweepstakes prizes are offered as at least 10% higher than the proportion of such buyers when no sweepstakes are on. Population 1: With Sweepstakes n1 = 300 H 0 : p1 p 2 0.10 H 1 : p1 p 2 0.10 x1 = 120 z p 1 = 0.40 Population 2: No Sweepstakes n 2 = 700 x 2 = 140 p 2 = 0.20 McGraw-Hill/Irwin ( p 1 p 2 ) D p (1 p ) 1 1 n1 p (1 p ) 2 2 n2 ( 0.40 0.20) 0.10 ( 0.40)( 0.60) ( 0.20)(.80) 700 300 Critical point: z 0.10 3.118 0.03207 = 3.09 0.001 H 0 may be rejected at any common level of significance. Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-10 BUSINESS STATISTICS 5th edi tion Example 8-9: Carrying Out the Test Standard Normal Distribution 0.4 f(z) 0.3 0.2 0.1 0.0 0 Nonrejection Region z z0.001=3.09 Rejection Region Test Statistic=3.118 McGraw-Hill/Irwin Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.001, the null hypothesis may be rejected, and we may conclude that the proportion of customers buying at least $2500 of travelers checks is at least 10% higher when sweepstakes are on. Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 8-11 5th edi tion Example 8-9: Using the Template P-value = 0.0009, so reject H0 at the 5% significance level. McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-12 BUSINESS STATISTICS 5th edi tion Confidence Intervals for the Difference between Two Population Proportions A (1-) 100% large-sample confidence interval for the difference between two population proportions: ( p1 p 2 ) z p (1 p ) 1 1 n1 p (1 p ) 2 2 n2 2 A 95% confidence interval using the data in example 8-9: p1 (1 p1 ) p 2 (1 p 2 ) ( 0.4 0.2) 1.96 ( 0.4 )( 0.6) ( 0.2)( 0.8) ( p1 p 2 ) z n2 300 700 n1 2 0.2 (1.96)( 0.0321) 0.2 0.063 [ 0.137 ,0.263] McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 8-13 5th edi tion Confidence Intervals for the Difference between Two Population Proportions – Using the Template – Using the Data from Example 8-9 McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-14 BUSINESS STATISTICS 5th edi tion 8-6 The F Distribution and a Test for Equality of Two Population Variances The F distribution is the distribution of the ratio of two chi-square random variables that are independent of each other, each of which is divided by its own degrees of freedom. An F random variable with k1 and k2 degrees of freedom: 12 F k ,k 1 2 k1 22 k2 McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-15 BUSINESS STATISTICS 5th edi tion The F Distribution McGraw-Hill/Irwin F Distributions with different Degrees of Freedom f(F) • The F random variable cannot be negative, so it is bound by zero on the left. • The F distribution is skewed to the right. • The F distribution is identified the number of degrees of freedom in the numerator, k1, and the number of degrees of freedom in the denominator, k2 . F(25,30) 1.0 F(10,15) 0.5 F(5,6) 0.0 Aczel/Sounderpandian 0 1 2 3 4 5 F © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-16 BUSINESS STATISTICS 5th edi tion Using the Table of the F Distribution Critical Points of the F Distribution Cutting Off a Right-Tail Area of 0.05 1 2 3 4 5 6 7 8 9 k2 1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 3.01 2.95 2.90 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 0.7 0.6 0.5 f(F) k1 F Distribution with 7 and 11 Degrees of Freedom 0.4 0.3 0.2 0.1 F 0.0 0 1 2 3 4 5 F0.05=3.01 The left-hand critical point to go along with F(k1,k2) is given by: 1 F k 2 ,k 1 Where F(k1,k2) is the right-hand critical point for an F random variable with the reverse number of degrees of freedom. McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-17 BUSINESS STATISTICS 5th edi tion Critical Points of the F Distribution: F(6, 9), = 0.10 F Distribution with 6 and 9 Degrees of Freedom 0.7 0.05 0.90 0.6 The right-hand critical point read directly from the table of the F distribution is: f(F) 0.5 F(6,9) =3.37 0.4 0.3 0.05 0.2 0.1 0.0 0 1 F0.95=(1/4.10)=0.2439 McGraw-Hill/Irwin 2 3 4 5 F The corresponding left-hand critical point is given by: 1 1 0.2439 F 9 , 6 410 . F0.05=3.37 Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-18 BUSINESS STATISTICS 5th edi tion Test Statistic for the Equality of Two Population Variances Test statistic for the equality of the variances of two normally distributed populations: F n 1,n 1 1 2 s12 2 s2 I: Two-Tailed Test 1 = 2 • H0: 1 = 2 • H1: 2 II: One-Tailed Test • • 12 • H0: 1 2 • H1: 1 2 McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-19 BUSINESS STATISTICS 5th edi tion Example 8-10 The economist wants to test whether or not the event (interceptions and prosecution of insider traders) has decreased the variance of prices of stocks. Population1 : Before n = 25 1 s 2 9.3 1 Population 2 : After n = 24 2 s 2 3.0 2 0.05 F 24,23 2.01 0.01 F 24,23 McGraw-Hill/Irwin H 0: H1: 2 1 2 1 2 2 21 2 2 s2 9.3 1 F F 3.1 3.0 n1 1, n 2 1 24,23 s2 2 H 0 may be rejected at a 1% level of significance. 2.70 Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-20 BUSINESS STATISTICS 5th edi tion Example 8-10: Solution Distribution with 24 and 23 Degrees of Freedom 0.7 0.6 f(F) 0.5 0.4 0.3 0.2 0.1 F 0.0 0 1 2 F0.01=2.7 McGraw-Hill/Irwin 3 4 5 Test Statistic=3.1 Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.01, the null hypothesis may be rejected, and we may conclude that the variance of stock prices is reduced after the interception and prosecution of inside traders. Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 8-21 5th edi tion Example 8-10: Solution Using the Template Observe that the pvalue for the test is 0.0042 which is less than 0.01. Thus the null hypothesis must be rejected at this level of significance of 0.01. McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 8-22 5th edi tion Example 8-11: Testing the Equality of Variances for Example 8-5 Population 1 Population 2 n = 14 1 2 2 s 0.12 1 0.05 F 13,8 3.28 0.10 F 13,8 2.50 McGraw-Hill/Irwin n =9 2 2 2 s 0.11 2 2 2 H : 0 1 2 2 2 H : 1 1 2 s2 0.12 2 1 F F 119 . 2 2 n 1 1 , n 2 1 13 , 8 s 0.11 2 H may not be rejected at the 10% level of significance. 0 Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE 8-23 BUSINESS STATISTICS 5th edi tion Example 8-11: Solution F Distribution with 13 and 8 Degrees of Freedom 0.7 0.10 0.80 0.6 f(F) 0.5 0.4 0.3 0.10 0.2 0.1 0.0 0 1 F0.90=(1/2.20)=0.4545 Test Statistic=1.19 McGraw-Hill/Irwin 2 3 4 F0.10=3.28 5 F Since the value of the test statistic is between the critical points, even for a 20% level of significance, we can not reject the null hypothesis. We conclude the two population variances are equal. Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 8-24 5th edi tion Template to test for the Difference between Two Population Variances: Example 8-11 Observe that the pvalue for the test is 0.8304 which is larger than 0.05. Thus the null hypothesis cannot be rejected at this level of significance of 0.05. That is, one can assume equal variance. McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 8-25 5th edi tion The F Distribution Template to McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 COMPLETE BUSINESS STATISTICS 8-26 5th edi tion The Template for Testing Equality of Variances McGraw-Hill/Irwin Aczel/Sounderpandian © The McGraw-Hill Companies, Inc., 2002 Penutup • Pembandingan Dua Populasi merupakan bagian dari pengujian Hipotesis dimana populasinya lebih dari satu,hal ini juga merupakan salah satu bentuk inferensial statistik yang berupa pengambilan kesimpulan/ pengambilan keputusan tentang menolak atau tidak menolak (menerima) suatu pernyataan/hipotesis 27