Suppose we observe $x$ and $y$ and we want to predict at $x=5$. A naive way would be to take each observation and compute $5/(x/y)$ or similarly $5*(y/x)$ and then take the overall mean. Thi is basically rescaling each observation to the unit scale and then extrapolating to 5.
A more sophisticated approach is to perform the linear regression and then predict at $x=5$.
Is there reason to believe the linear regression approach is more accurate compared to the first method? I believe the first approach is very sample dependent.
Here is an example:
library(ggplot2) set.seed(123) nobs=1000 x=runif(nobs,0.1,100) y=abs(x*.05+rnorm(nobs,0,1)) a2=data.frame(x,y) ggplot(a2,aes(x=x,y=y))+geom_point()+geom_smooth()+geom_smooth(method='lm') ### Linear Regression Prediction fit=lm(y~x) print(predict(fit,newdata=data.frame(x=100),interval='confidence')) # fit lwr upr # 1 4.844763 4.729187 4.960339 ### Naive Prediction print(mean(100/(a2$x/a2$y))) # 8.49