###Simple Linear Regression Model using R
###UNIFE 
###Spring Semester
###Mini V. 27-02-2019

#recall the prev analysis: 
#we already performed our regression analysis 1-6
getwd()
cake=read.csv2("cake_reg lin.csv")
attach(cake)
View(cake)
y=sold_cakes
x=unit_price
reg_lin=lm(y~x)




#TOPIC:  the strength of the correlation between x and y within the population 
#we test the system of hypothesis based on the correlation coefficient RHO (?) #

#H0: RHO=0 (no correlation)
#H1: RHO dif. from 0 (correlazione –with given intensity)

#if t-statr > v.c. --> we reject H0 --> thus, it exists a correlation –with a given intensity – within the reality between x and y

#tstat.r=(r-RHO)/root.sq of ((1-r^2)/n-2))    c.v.:  a/2  and d.f = n-2

dev.tot=sum((y-mean(y))^2)                  #total residuals SST
dev.disp=sum(reg_lin$residuals^2)           #residuals SSE
dev.reg=dev.tot-dev.disp                    #regression’s residuals SSR

r=sqrt(dev.reg/dev.tot)
r #there is a "moderate/strong" positive correlation

num=r #numerator of tstat.r

den=sqrt((1-(dev.reg/dev.tot))/32) # denominator of tstat.r

tstat.r=num/den

tstat.r #5.208465

vc.r=qt(0.025,32)

vc.r #-2.037

# (ABS VALUE!): tstat.r>C.V.r --> we reject H0 --> it exists a correlation –whit that given strenght- within the reality 
# significant correlation between  x and  y #