a <- c(1.186,1.151,1.322,1.339,1.200,1.402,1.365,1.537,1.559)
b <- c(1.061,0.992,1.063,1.062,1.065,1.178,1.037,1.086,1.052)
c <- (a-b)
if the mean of the two samples is the same, then the difference in the two means is zero. Null hypothesis.meanA-meanB=0. The alternate hypothesis is that the difference in means is not zero. meanA-meanB!=0
library(dplyr)
##
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
##
## filter, lag
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
qqnorm(a)
qqline(a)
qqnorm(b)
qqline(b)
qqnorm(c)
qqline(c)
boxplot(a,b,col=c("light blue","red"),names=c("Karlsruhe Method","Lehigh Method"))
the result of the box plot clearly says the variance are not equal.
alternative hypothesis is tested
wilcox.test(a,b)
##
## Wilcoxon rank sum exact test
##
## data: a and b
## W = 80, p-value = 8.227e-05
## alternative hypothesis: true location shift is not equal to 0
t.test(a,b)
##
## Welch Two Sample t-test
##
## data: a and b
## t = 5.3302, df = 9.8059, p-value = 0.0003557
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.1590886 0.3886892
## sample estimates:
## mean of x mean of y
## 1.340111 1.066222
t.test(a,b,paired=TRUE)
##
## Paired t-test
##
## data: a and b
## t = 6.0819, df = 8, p-value = 0.0002953
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.1700423 0.3777355
## sample estimates:
## mean of the differences
## 0.2738889
the p-value = 8.227e-05 whis is really less than alfa!=0.05 so, we reject null hypothesis.
we have a clear evidence that we can conclude the mean values are not equal.
95 percent confidence interval:0.1590886 0.3886892
null hypothesis is H0 : µ1 = µ2
alternate hypothesis H1: µ1 != µ2
a <- c(1.186,1.151,1.322,1.339,1.200,1.402,1.365,1.537,1.559)
b <- c(1.061,0.992,1.063,1.062,1.065,1.178,1.037,1.086,1.052)
c <- (a-b)
qqnorm(a)
qqline(a)
qqnorm(b)
qqline(b)
qqnorm(c)
qqline(c)
boxplot(a,b,col=c("light blue","red"),names=c("Karlsruhe Method","Lehigh Method"))
wilcox.test(a,b)
##
## Wilcoxon rank sum exact test
##
## data: a and b
## W = 80, p-value = 8.227e-05
## alternative hypothesis: true location shift is not equal to 0
t.test(a,b)
##
## Welch Two Sample t-test
##
## data: a and b
## t = 5.3302, df = 9.8059, p-value = 0.0003557
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.1590886 0.3886892
## sample estimates:
## mean of x mean of y
## 1.340111 1.066222
t.test(a,b,paired=TRUE)
##
## Paired t-test
##
## data: a and b
## t = 6.0819, df = 8, p-value = 0.0002953
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 0.1700423 0.3777355
## sample estimates:
## mean of the differences
## 0.2738889