###################################################################################
#
# O objetivo deste problema é investigar os determinantes da 
# performance média dos alunos de diferentes escolas (n = 420) 
# da Califórnia em um teste padronizado aplicado aos alunos de
# quinta série (TESTSCR). As informações de interesse se encontram 
# no banco de dados caliescom.xls.
#
# STR     – número de alunos / número de professores;
# MEALPCT – porcentagem de alunos que se qualificam para o lanche subsidiado.
#
###################################################################################
rm(list=ls())
nomes = c("Nota no teste","razao alunos por professor","% lanche subsidiado")
data  = read.csv("caliescom.csv",header=TRUE)
n     = nrow(data)
data[,1] = 100*data[,1]/max(data[,1])
attach(data)

par(mfrow=c(1,2))
for (i in 2:3){
  plot(data[,i],data[,1],xlab=nomes[i],ylab=nomes[1])
  abline(lm(data[,1]~data[,i]),col=2,lwd=2)
}

reg = lm(TESTSCR~STR+MEALPCT)
summary(reg) 
yhat = reg$fit 
e    = reg$res
s2   = sum(e^2)/(n-3)
s    = sqrt(s2)
cov  = round(100*mean(abs(e)<2*s),1)

par(mfrow=c(1,2))
plot(yhat,e,xlab="Fitted",ylab="Residuals",pch=16)
abline(h=0,lty=2)
abline(h=-2*s,lty=2)
abline(h=2*s,lty=2)
title(paste("Coverage = ",cov,"%",sep=""))
hist(e,prob=TRUE,xlab="Residuals",main="");box()
xxx = seq(min(e),max(e),length=1000)
lines(xxx,dnorm(xxx,0,s),col=2,lwd=2)




# RESET test
reg1  = lm(TESTSCR~STR+MEALPCT)
yhat2 = reg1$fit^2 
yhat3 = reg1$fit^3
reg2  = lm(TESTSCR~STR+MEALPCT+yhat2+yhat3)

summary(reg2)