For making prediction intervals, the book (Applied statistics and probability for engineers by montgomery and runger 5th edition) creates a new random variable called prediction error and based on that (after translation and scaling to standard $T$) it creates a PI for $X_{n+1}$. My question is this: Why not use $\overline{X}$ as an estimate for mean of $X_{n+1}$ and use sample standard dev as estimate for std of $X_{n+1}$ and basically create a PI based on that like we do for confidence interval. Why use prediction error?
What is wrong with my method?
Thank you
