Untuk pengujian dua nilai tengah populasi dikatakan berpasangan jika pengambilan unit-unit contoh pertama memperhatikan bagaimana unit-unit contoh kedua dipilih. Untuk melihat perbedaan dua populasi dari kasus dua contoh berpasangan dapat dilakukan dengan secara langsung membedakan setiap obyek pada contoh satu dan contoh dua untuk setiap pasangan.
Jika dimisalkan beda nilai tengah populasi dinotasikan dengan \(\mu_d = \mu_1 - \mu_2\), maka penduga tak bias dari \(\mu_d\) adalah nilai tengah dari beda dua contoh \((\bar{d})\), yang diperoleh dari: \[\bar{d}=\frac{\sum_{i=1}^nd_i}{n}\] dengan galat baku: \[s_d=\sqrt{\frac{\sum_{i=1}^nd_i}{n}-\frac{n}{n-1}\bar{d} ^2}\] pengujian beda nilai tengah populasi, bentuk hipotesisnya dibedakan menjadi tiga, yaitu:
1. \(H_0 : \delta = \delta_0\) vs \(H_1 : \delta < \delta_0\) (Uji Satu Arah)
2. \(H_0 : \delta = \delta_0\) vs \(H_1 : \delta > \delta_0\) (Uji Satu Arah)
3. \(H_0 : \delta = \delta_0\) vs \(H_1 : \delta \ne \delta_0\) (Uji Dua Arah)
sehingga statistik uji yang digunakan adalah t-student sebagai berikut: \[t_{hitung}=\frac{(\bar{d}-\delta_0)\sqrt{n}}{s_d}\] dengan derajat bebas (db) sebesar \(n-1\)
Import Data
x1 <- c(58.4, 60.3, 61.7, 69.2, 64.0, 62.6, 56.7)
x2 <- c(60.0, 54.8, 58.1, 62.1, 58.5, 59.9, 54.4)\(H_{0} : \mu_1 - \mu_2 = 0\) atau \(\delta = 0\)
\(H_{1} : \mu_1 - \mu_2 \ne 0\) atau \(\delta \ne 0\)
t.test(x1, x2, alternative = "two.sided")##
## Welch Two Sample t-test
##
## data: x1 and x2
## t = 1.9136, df = 10.644, p-value = 0.08293
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.5554876 7.7269162
## sample estimates:
## mean of x mean of y
## 61.84286 58.25714\(H_{0} : \mu_1 - \mu_2 = 0\) atau \(\delta = 0\)
\(H_{1} : \mu_1 - \mu_2 > 0\) atau \(\delta > 0\)
t.test(x1, x2, alternative = "greater")##
## Welch Two Sample t-test
##
## data: x1 and x2
## t = 1.9136, df = 10.644, p-value = 0.04147
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
## 0.2101727 Inf
## sample estimates:
## mean of x mean of y
## 61.84286 58.25714\(H_{0} : \mu_1 - \mu_2 = 0\) atau \(\delta = 0\)
\(H_{1} : \mu_1 - \mu_2 < 0\) atau \(\delta < 0\)
t.test(x1, x2, alternative = "less")##
## Welch Two Sample t-test
##
## data: x1 and x2
## t = 1.9136, df = 10.644, p-value = 0.9585
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
## -Inf 6.961256
## sample estimates:
## mean of x mean of y
## 61.84286 58.25714\(H_{0} : \mu_1-\mu_2 = 4.5\) atau \(\delta = 4.5\)
\(H_{1} : \mu_1-\mu_2 \ne 4.5\) atau \(\delta \ne 4.5\)
t.test(x1, x2, alternative = "two.sided", mu = 4.5)##
## Welch Two Sample t-test
##
## data: x1 and x2
## t = -0.48792, df = 10.644, p-value = 0.6355
## alternative hypothesis: true difference in means is not equal to 4.5
## 95 percent confidence interval:
## -0.5554876 7.7269162
## sample estimates:
## mean of x mean of y
## 61.84286 58.25714\(H_{0} : \mu_1-\mu_2 = 4.5\) atau \(\delta = 4.5\)
\(H_{1} : \mu_1-\mu_2 > 4.5\) atau \(\delta > 4.5\)
t.test(x1, x2, alternative = "greater", mu = 4.5)##
## Welch Two Sample t-test
##
## data: x1 and x2
## t = -0.48792, df = 10.644, p-value = 0.6822
## alternative hypothesis: true difference in means is greater than 4.5
## 95 percent confidence interval:
## 0.2101727 Inf
## sample estimates:
## mean of x mean of y
## 61.84286 58.25714\(H_{0} : \mu_1-\mu_2 = 4.5\) atau \(\delta = 4.5\)
\(H_{1} : \mu_1-\mu_2 < 4.5\) atau \(\delta < 4.5\)
t.test(x1, x2, alternative = "less", mu = 4.5)##
## Welch Two Sample t-test
##
## data: x1 and x2
## t = -0.48792, df = 10.644, p-value = 0.3178
## alternative hypothesis: true difference in means is less than 4.5
## 95 percent confidence interval:
## -Inf 6.961256
## sample estimates:
## mean of x mean of y
## 61.84286 58.25714