#Here we use our age data to explore the "danger of averaging".  

# This vector (called A) contains the actual reported Ages of our 10 subjects in years.

A=c(35,44,48,42,23,44,54,22,54,26)

# This array contains the guesses made by our 9 in class groups.  The rows are the rows samples and the columns are the group.  So G[i,j] is the person i's age as guessed by group j.   


G=cbind(
c(36,30,37,32,25,31,62,23,43,24),
c(27,32,43,37,24,35,56,22,43,23),
c(29,38,35,36,22,30,55,24,43,22),
c(26,27,36,32,28,35,58,24,43,27),
c(29,25,41,22,19,33,58,24.5,38,22),
c(32,30,37,28,28,34,56,26,42,25),
c(35,40,55,36,30,35,60,27,41,22),
c(26,30,43,21,21,33,58,18,42,17),
c(35,28,41,33,29,32,60,27,43,24)
)

# First let us compare with a scatter plot the actual Ages with the average of our group Ages, basically the guess we would make as a class.

# First let us compare with a scatter plot the actual ages with the average of our group ages, basically the guess we would make as a class. We let mMG denote the mean guess for a given picture over all 9 of our groups.

MG=apply(G,1,mean)

PA=(2/51)*A^2-(7/3)*A+1000/17
plot(PA,MG,main="Comparing  Guessed Ages with Perceived Ages",xlab=" Our Model of Perceived Ages",ylab="Class's Consensus Estimated Age")


# Here they are again.  Play with outliers.  How does changing 58.1111 effect the slope and correlation coefficient?  Put in some outliers explore.  


MGf=c(30.55556, 31.11111, 40.88889, 30.77778, 25.11111, 33.11111, 58.11111, 23.94444, 42.00000, 22.88889)


PAf=c(25.19608, 32.07843, 37.17647, 30.00000, 25.90196, 32.07843, 47.17647, 26.47059, 47.17647, 24.66667)

plot(PAf,MGf)

cor(PAf,MGf)

lm(PAf ~ MGf)

abline(lm(PAf ~ MGf),col="red")

quartz()


######

# We know look at all the data.  How should this effect the correlation?  WHow should this affect our interpretation of the results? 

Aall=rep(A,9)
Gall<-G
dim(Gall)<-c(length(Gall),1)

plot(Aall,Gall,main="Comparing  Guessed Ages with Perceived Ages",xlab=" Actual  Ages",ylab="Each Group's Estimated Age")

cor(Aall,Gall)


PAall=(2/51)*Aall^2-(7/3)*Aall+1000/17


plot(PAall,Gall,main="Comparing  Guessed Ages with Perceived Ages",xlab=" Our Model of Perceived Ages",ylab="Each Group's Estimated Age")

cor(PAall,Gall)


rAPAall=cor(PAall,Gall)
mPAall=mean(PAall)
sPAall=sd(PAall)
mGall=mean(Gall)
sGall=sd(Gall)
b1=(sGall/sPAall)*rAPAall
b0=mGall-b1*(mPAall)

hatGall=b1*PAall+b0


plot(PAall,Gall,main="Comparing  Guessed Ages with Perceived Ages",xlab=" Our Model of Perceived Ages",ylab="Each Group's Estimated Age")
points(PAall,hatGall,col="red")

# Question 1.  lm(X ~ Y)  should be read as "please produce a linear model to predict Guesses from FofAges."  Explain what it actually does.

lm(Gall ~ PAall)

ResAll=Gall-hatGall

hist(ResAll)
plot(PAall,ResAll)