Suppose we have the following data from two independent samples:
Sample 1 20.0 12.4 23.6 19.5 33.5 11.7 13.6 18.2 11.8 21.8
Sample 2 16.8 21.7 31.5 22.1 23.8 17.1 20.0 17.1
First, read the sample into two vectors x1 and x2:
x1 <- c(20.0, 12.4, 23.6, 19.5, 33.5, 11.7, 13.6, 18.2, 11.8, 21.8)
x2 <- c(16.8, 21.7, 31.5, 22.1, 23.8, 17.1, 20.0, 17.1)To test the following hypotheses
\[ H_0 : \mu_1 − \mu_2 = 2 \\ H_a : \mu_1 − \mu_2 < 2 \]
using the equal variances test use the following R command:
t.test(x1, x2, mu = 2, alternative = "less", var.equal = T)##
## Two Sample t-test
##
## data: x1 and x2
## t = -1.621, df = 16, p-value = 0.06227
## alternative hypothesis: true difference in means is less than 2
## 95 percent confidence interval:
## -Inf 2.358326
## sample estimates:
## mean of x mean of y
## 18.6100 21.2625or classical T test
Two Sample t-test
To test the following hypotheses
\[ H_0 : \mu_1 − \mu_2 = 0 \\ H_a : \mu_1 − \mu_2 = 0 \] using the unequal variances test use the following R command:
t.test(x1, x2, mu = 0, alternative = "two.sided")##
## Welch Two Sample t-test
##
## data: x1 and x2
## t = -0.9594, df = 15.873, p-value = 0.3517
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -8.517329 3.212329
## sample estimates:
## mean of x mean of y
## 18.6100 21.2625Note: Using alternative = "two.sided" also provides a
confidence interval. To change the confidence level to 99%, add the
option conf.level = 0.99.