Statistik Bisnis 2 Week 5 Comparing the Means of Two Independent Populations
Learning Objectives The means of two independent populations
The means of two related populations
In this chapter, you learn how to use hypothesis testing for comparing the difference between:
The proportions of two independent populations
The variances of two independent populations by testing the ratio of the two variances
Two-Sample Tests Two-Sample Tests
Population Means
Independent Samples
unknown, assumed equal
unknown, assumed unequal
Population Proportions
Related Samples
Population Variances
Difference Between Two Means Independent Samples unknown, assumed equal unknown, assumed unequal
Goal: Test hypothesis or form a confidence interval for the difference between two population means, μ1 – μ2 The point estimate for the difference is
X1 – X2
Difference Between Two Means: Independent Samples Different data sources • Unrelated Independent • Independent
Samples
unknown, assumed equal unknown, assumed unequal
– Sample selected from one population has no effect on the sample selected from the other population Use Sp to estimate unknown σ. Use a Pooled-Variance t test.
Use S1 and S2 to estimate unknown σ1 and σ2. Use a Separate-Variance t test.
Hypothesis Tests for Two Population Means Two Population Means, Independent Samples
Lower-tail test:
Upper-tail test:
Two-tail test:
H0: μ1 μ2 H1: μ1 < μ2
H0: μ1 ≤ μ2 H1: μ1 > μ2
H0: μ1 = μ2 H1: μ1 ≠ μ2
i.e.,
i.e.,
i.e.,
H0: μ1 – μ2 0 H1: μ1 – μ2 < 0
H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0
H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0
Hypothesis tests for μ1 – μ2 Two Population Means, Independent Samples Lower-tail test:
Upper-tail test:
Two-tail test:
H0: μ1 – μ2 0 H1: μ1 – μ2 < 0
H0: μ1 – μ2 ≤ 0 H1: μ1 – μ2 > 0
H0: μ1 – μ2 = 0 H1: μ1 – μ2 ≠ 0
a
a -ta
Reject H0 if tSTAT < -ta
ta Reject H0 if tSTAT > ta
a/2 -ta/2
a/2 ta/2
Reject H0 if tSTAT < -ta/2 or tSTAT > ta/2
Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and assumed equal Independent Samples unknown, assumed equal unknown, assumed unequal
Assumptions: Samples are randomly and independently drawn Populations are normally distributed or both sample sizes are at least 30 Population variances are unknown but assumed equal
Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and assumed equal (continued) • The pooled variance is:
Independent Samples unknown, assumed equal unknown, assumed unequal
2 2 n 1 S n 1 S 1 2 2 S2 1 p
(n1 1) (n 2 1)
• The test statistic is:
t STAT
X
1
X 2 μ1 μ 2 1 1 S n1 n 2 2 p
• Where tSTAT has d.f. = (n1 + n2 – 2)
Confidence interval for µ1 - µ2 with σ1 and σ2 unknown and assumed equal Independent Samples unknown, assumed equal unknown, assumed unequal
The confidence interval for μ1 – μ2 is:
X
1
X 2 ta/2
1 1 S n1 n 2
Where tα/2 has d.f. = n1 + n2 – 2
2 p
Pooled-Variance t Test Example You are a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQ Number 21 25 Sample mean 3.27 2.53 Sample std dev 1.30 1.16 Assuming both populations are approximately normal with equal variances, is there a difference in mean yield (a = 0.05)?
Pooled-Variance t Test Example: Calculating the Test Statistic (continued) H0: μ1 - μ2 = 0 i.e. (μ1 = μ2) H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2) The test statistic is:
t STAT
X
1
X 2 μ1 μ 2 1 1 S n1 n 2 2 p
3.27 2.53 0 1 1 1.5021 21 25
2 2 2 2 n 1 S n 1 S 21 1 1.30 25 1 1.16 1 2 2 S2 1 P
(n1 1) (n 2 1)
(21 - 1) ( 25 1)
2.040
1.5021
Pooled-Variance t Test Example: Hypothesis Test Solution H0: μ1 - μ2 = 0 i.e. (μ1 = μ2) H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2) a = 0.05 df = 21 + 25 - 2 = 44 Critical Values: t = ± 2.0154
Reject H0
.025
-2.0154
Reject H0
.025
0 2.0154
t
2.040
Test Statistic: Decision: 3.27 2.53 t STAT 2.040 Reject H0 at a = 0.05 1 1 Conclusion: 1.5021 21 25 There is evidence of a
difference in means.
Pooled-Variance t Test Example: Confidence Interval for µ1 - µ2 DCOVA
Since we rejected H0 can we be 95% confident that µNYSE > µNASDAQ? 95% Confidence Interval for µNYSE - µNASDAQ
X X t 1
2
a/2
1 1 S 0.74 2.0154 0.3628 (0.009, 1.471) n1 n 2 2 p
Since 0 is less than the entire interval, we can be 95% confident that µNYSE > µNASDAQ
Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown, not assumed equal Independent Samples unknown, assumed equal unknown, assumed unequal
Assumptions: Samples are randomly and independently drawn Populations are normally distributed or both sample sizes are at least 30 Population variances are unknown and cannot be assumed to be equal
Hypothesis tests for µ1 - µ2 with σ1 and σ2 unknown and not assumed equal (continued) The test statistic is:
Independent Samples unknown, assumed equal
t STAT
X
1
X 2 μ1 μ 2 S12 S 22 n1 n 2
tSTAT has d.f. ν =
unknown, assumed unequal
2
S1 S 2 n n 1 2 2 2 2 2 S1 S2 n 1 n2 n1 1 n2 1 2
2
EXERCISE
10.7 Menurut sebuah penelitian baru-baru ini, ketika berbelanja barang-barang mewah dalam jaringan (online shopping), pria rata-rata membaelanjakan $2,401, sementara wanita rata-ratanya $1,527. Misalkan penelitian tersebut dilakukan pada 600 orang pria dan 700 orang wanita, dan simpangan baku dari jumlah uang yang dibelanjakan tersebut adalah $1,200 untuk pria dan $1,000 untuk wanita. a. Tentukan hipothesis kosong dan alternatifnya jika anda ingin menentukan apakah rata-rata uang yang dibelanjakan oleh pria lebih banyak daripada wanita b. Pada konteks penelitian ini, apakah yang dimaksud dengan kesalahan tipe I? c. Pada konteks penelitian ini, apakah yang dimaksud dengan kesalahan tipe I? d. Dengan tingkat signifikansi 0.01, apakah terdapat bukti bahwa rata-rata jumlah uang yang dibelanjakan pria lebih banyak daripada wanita?
10.10 (1) Computer Anxiety Rating Scale (CARS) mengukur tingkat kecemasan terhadap komputer (computer anxiety), dengan skala dari 20 (tidak ada kecemasan) hingga 100 (sangat cemas). Peneliti dari Miami University menyebarkan CARS pada 172 mahasiswa bisnis. Salah satu tujuan dari penelitian tersebut adalah menentukan apakah terdapat perbedaan tingkat kecemasan komputer yang dirasakan oleh mahasiswa bisnis pria dan wanita. Mereka menemukan data berikut: X S n
Pria 40,26 13,35 100
Wanita 36,85 9,42 72
10.10 (2) a. Dengan tingkat signifikansi 0.05, apakah terdapat bukti bahwa kecemasan komputer yang dirasakan oleh mahasiswa bisnis wanita berbeda dari yang dirasakan oleh mahasiswa bisnis pria? b. Apakah asumsi-asumsi yang harus anda buat mengenai kedua populasi tersebut untuk dapat menggunakan uji t?
10.16 (1) Apakah anak-anak menggunakan telepon selular? Sepertinya demikian, menurut penelitian baru-baru ini, pengguna telepon selular berusia dibawah 12 tahun ratarata melakukan 137 panggilan telepon per bulan. Cukup tinggi, jika dibandingkan dengan 231 panggilan telepon per bulan yang dilakukan oleh pengguna telepon selular berusia 13 hingga 17 tahun. Misalkan hasil tersebut diambil dari sampel 50 orang pengguna telepon selular untuk setiap grup pengguna dan simpangan baku sampel pengguna telepon selular berusia dibawah 12 tahun adalah 51,7 panggilan telepon per bulan dan simpangan baku sampel pengguna telepon selular berusia 13 hingga 17 tahun adalah 67,6 panggilan telepon per bulan.
10.16 (2) a. Dengan mengasumsikan bahwa variansi populasi dari pengguna telepon selular adalah sama, adakah bukti yang menunjukkan bahwa terdapat perbedaan rata-rata penggunaan telepon selular antara kelompok usia dibawah 12 tahun dan kepompok usia 13 hingga 17 tahun? (Gunakan tingkat signifikansi 0,05.) b. Selain kesamaan variansi, sebutkan asumsi lain yang diperlukan dalam melakukan uji hipotesis pada poin (a)?
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