#Paired t-test:

x<-read.csv(file="C:/Rdata/co2.csv")
x

attach(x)
X1=January
X2=December
d=X1-X2
n=length(d)
n
#n=39
dbar=mean(d)
dbar
#dbar=-0.2638462
sd=sqrt(var(d))
sd
#sd=0.500482


#Performing the Paired t-test:

t.test(X1,X2,alternative="two.sided",paired=T,conf.level=0.95)

#Critical Value:

n=39
a=0.05
C1=qt(a/2,n-1)
C1    <- Two-sided lower critical value
C2=qt(1-a/2,n-1)
C2    <- Two-sided upper critical value
OR
C1=qt(a/2,n-1)
C2=abs(C1)    <- C2 is the absolute value of C1


#Probability Value

#t < 0

t=-3.2923
P=2*pt(t,n-1)
P


Wilcoxon Signed-Rank Test:

x<-read.csv(file="C:/Rdata/co2(d).csv")
x
attach(x)

#Performing the Wilcoxon Signed-Rank Test:

wilcox.test(January,December,alternative="two.sided",paired=T)

#Critical Value of the Test

a=0.05
C=qnorm(1-a/2,0,1)
C

Sign Test:

x<-read.csv(file="C:/Rdata/co2(s).csv")
x
attach(x)

n=length(January)
n
#n=39
p=0.5
PLUS=length(January[d=="plus"])
PLUS
#PLUS=12
MINUS=length(January[d=="minus"])
MINUS
#MINUS=27

#Performing the Sign Test:

binom.test(PLUS,n,p, alternative="two.sided",conf.level=0.95)

#Alternative calculation using the Binomial function pbinom():

Since C=12 and n/2=19.5,
C < n/2

C=12
p=0.5
P=2*pbinom(C,n,p)