x<-c(2,3,4,5,6)
y<-c(4,4,6,6,10)
x;y

(n<-length(x))

(sumx<-sum(x))
(sumy<-sum(y))
(mx=mean(x))
(my=mean(y))
(xy<-x*y)
(sumxy<-sum(xy))
(sumxsq<-sum(x^2))
(sumysq<-sum(y^2))

beta1<-(sumxy-mx*sumy)/(sumxsq-mx*sumx)
beta0<-my-beta1*mx
beta0;beta1

(b1hat<-cov(x,y)/var(x))
(b0hat<-mean(y)-b1hat*mean(x))

(dx<-x-mx)
(dy<-y-my)
(sumdxsq<-sum(dx^2))
(sumdysq<-sum(dy^2))
(sumdxdy<-sum(dx*dy))
(ssr<-sumdysq-beta1*sumdxdy)
(sigusq<-ssr/3)

(vbeta1<-sigusq/sumdxsq)

(rsq<-(beta1^2*sumdxsq)/sumdysq)

(t<-beta1/sqrt(vbeta1))
(pt(t,3))

(tc<-qt(p=0.025, df=3, lower.tail=F))

(b1hat_lb<-b1hat-(tc)*sqrt(vbeta1))
(b1hat_ub<-b1hat+(tc)*sqrt(vbeta1))

x0<-7
(yhat<-beta0+beta1*x0)

(sigesq_ind<-sigusq*(1+(1/n)+((x0-mx)^2/sumdxsq)))
(sige_ind<-sqrt(sigesq_ind))
(yhat_ind_lb<-(yhat-(tc)*sige_ind))
(yhat_ind_ub<-(yhat+(tc)*sige_ind))


(sigesq_mean<-sigusq*((1/n)+((x0-mx)^2/sumdxsq)))
(sige_mean<-sqrt(sigesq_mean))
(yhat_mean_lb<-(yhat-(tc)*sige_mean))
(yhat_mean_ub<-(yhat+(tc)*sige_mean))