Statistik Bisnis 2. Week 6 Two-Sample Test Population Proportions and Variances

Transkripsi

1 Statistik Bisnis Week 6 Two-Samle Test Poulation Proortions and Variances

2 Learning Objectives The means of two indeendent oulations The means of two related oulations In this chater, you learn how to use hyothesis testing for comaring the difference between: The roortions of two indeendent oulations The variances of two indeendent oulations by testing the ratio of the two variances

3 Two-Samle Tests Two-Samle Tests Poulation Means Poulation Proortions Poulation Variances Indeendent Samles Related Samles unknown, assumed equal unknown, assumed unequal

4 Two Poulation Proortions Poulation roortions Goal: test a hyothesis or form a confidence interval for the difference between two oulation roortions, π π The oint estimate for the difference is

5 Two Poulation Proortions Poulation roortions In the null hyothesis we assume the null hyothesis is true, so we assume π = π and ool the two samle estimates The ooled estimate for the overall roortion is: X n X n where X and X are the number of items of interest in samles and

6 Two Poulation Proortions ) ( n n π π Z STAT The test statistic for π π is a Z statistic: (continued) n X, n X, n n X X where Poulation roortions

7 Hyothesis Tests for Two Poulation Proortions Poulation roortions Lower-tail test: Uer-tail test: Two-tail test: H 0 : π π H : π < π i.e., H 0 : π π 0 H : π π < 0 H 0 : π π H : π > π i.e., H 0 : π π 0 H : π π > 0 H 0 : π = π H : π π i.e., H 0 : π π = 0 H : π π 0

8 Hyothesis Tests for Two Poulation Proortions (continued) Lower-tail test: H 0 : π π 0 H : π π < 0 Poulation roortions Uer-tail test: H 0 : π π 0 H : π π > 0 Two-tail test: H 0 : π π = 0 H : π π 0 a a a/ a/ -z a z a -z a/ z a/ Reject H 0 if Z STAT < -Z a Reject H 0 if Z STAT > Z a Reject H 0 if Z STAT < -Z a/ or Z STAT > Z a/

9 Hyothesis Test Examle: Two oulation Proortions Is there a significant difference between the roortion of men and the roortion of women who will vote Yes on Proosition A? In a random samle, 36 of 7 men and 35 of 50 women indicated they would vote Yes Test at the.05 level of significance

10 Hyothesis Test Examle: Two oulation Proortions The hyothesis test is: H 0 : π π = 0 (the two roortions are equal) H : π π 0 (there is a significant difference between roortions) The samle roortions are: Men: = 36/7 = 0.50 Women: = 35/50 = 0.70 The ooled estimate for the overall roortion is: X n X n (continued)

11 Hyothesis Test Examle: Two oulation Proortions (continued) Reject H 0 Reject H 0 The test statistic for π π is: z STAT π π ( ) n (. 58 ) n 7 Critical Values = ±.96 For a = Decision: Reject H 0.05 Conclusion: There is evidence of a difference in roortions who will vote yes between men and women.

12 Confidence Interval for Two Poulation Proortions Poulation roortions The confidence interval for π π is: Z a/ ( n ) ( n )

13 Testing for the Ratio Of Two Poulation Variances Hyotheses F STAT Tests for Two Poulation Variances F test statistic H 0 : σ = σ H : σ σ H 0 : σ σ H : σ > σ Where: S = Variance of samle (the larger samle variance) n = samle size of samle S = Variance of samle (the smaller samle variance) n = samle size of samle n = numerator degrees of freedom n = denominator degrees of freedom S S

14 The F Distribution The F critical value is found from the F table There are two degrees of freedom required: numerator and denominator The larger samle variance is always the numerator When F STAT In the F table, S S df = n ; df = n numerator degrees of freedom determine the column denominator degrees of freedom determine the row

15 Finding the Rejection Region H 0 : σ = σ H : σ σ H 0 : σ σ H : σ > σ a/ F a 0 Do not reject H 0 F α/ Reject H 0 0 Do not reject H 0 F α Reject H 0 F Reject H 0 if F STAT > F α/ Reject H 0 if F STAT > F α

16 F Test: An Examle You are a financial analyst for a brokerage firm. You want to comare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data: NYSE NASDAQ Number 5 Mean Std dev.30.6 Is there a difference in the variances between the NYSE & NASDAQ at the a = 0.05 level?

17 F Test: Examle Solution Form the hyothesis test: H 0 : σ H : σ σ σ Find the F critical value for a = 0.05: (there is no difference between variances) (there is a difference between variances) Numerator d.f. = n = = 0 Denominator d.f. = n = 5 = 4 F α/ = F.05, 0, 4 =.33

18 F Test: Examle Solution The test statistic is: (continued) H 0 : σ = σ H : σ σ S.30 F STAT.56 S.6 a/ =.05 F STAT =.56 is not in the rejection region, so we do not reject H 0 0 Do not reject H 0 Reject H 0 F 0.05 =.33 F Conclusion: There is insufficient evidence of a difference in variances at a =.05

19 EXERCISE

20 0.30 () Aakah tahun ini dierlukan usaha lebih untuk keluar dari sebuah list dari tahun sebelumnya? Sebuah enelitian dari 00 online retailer besar menunjukkan data berikut: PERLU TIGA ATAU LEBIH KLIK SEBELUM KELUAR TAHUN Ya Tidak

21 0.30 () a. Tentukan hiothesis kosong dan hiothesis alternatif untuk mengetahui aakah dierlukan usaha lebih untuk keluar dari sebuah list jika dibandingkan dengan tahun sebelumnya. b. Lakukan uji hiothesis untuk oin (a), dengan menggunakan tingkat signifikansi 0,05. c. Aakah hasil dari oin (b) sesuai dengan klaim bahwa dierlukan usaha lebih untuk keluar dari sebuah list jika dibandingkan dengan tahun sebelumnya?

22 0.34 Bagaimana erasaan orang Amerika mengenai iklan di halaman web? Sebuah survey yang dilakukan ada.000 engguna internet dewasa diketahui bahwa 670 orang menentang adanya iklan di halaman web. Misalkan sebuah survei lain ada.000 orang engguna internet berusia 7 tahun menemukan bahwa 50 orang menentang adanya iklan di halaman web. a. Dengan menggunakan tingkat signifikansi 0,05, adakah bukti bahwa terdaat erbedaan roorsi antara engguna internet dewasa dan engguna internet berusia 7 tahun yang menentang iklan? b. Temukan -value ada oin (a) dan interretasikan maksudnya.

23 0.46 () Comuter Anxiety Rating Scale (CARS) mengukur tingkat kecemasan terhada komuter (comuter anxiety), dengan skala dari 0 (tidak ada kecemasan) hingga 00 (sangat cemas). Peneliti dari Miami University menyebarjab CARS ada 7 mahasiswa bisnis. Salah satu tujuan dari enelitian tersebut adalah menentukan aakah terdaat erbedaan tingkat kecemasan komuter yang dirasakan oleh mahasiswa bisnis ria dan wanita. Mereka menemukan data berikut: Pria Wanita X S n 00 7

24 0.46 () a. Dengan tingkat signifikansi 0,05, adakah bukti yang menunjukkan erbedaan sebaran (variability) kecemasan komuter (comuter anxiety) yang dialami ria dan wanita? b. Asumsi aa yang anda erlukan tentang kedua oulasi tersebut untuk daat menggunakan uji F? c. Berdasarkan oin (a) dan (b), uji t manakah yang seharusnya anda gunakan untuk menguji aakah ada erbedaan yang signifikan antara kecemasan komuter yang dirasakan oleh wanita dan ria?

25 THANK YOU