Here is what my notes say about estimation and prediction:

Estimating the conditional mean

We need to estimate the conditional mean $\beta_0+\beta_1x_0$ at a value $x_0$, so we use $\hat{Y_0}=\hat{\beta_0}+\hat{\beta_1}x_0$ as a natural estimator.

Here we get $$ \hat{Y_0} \sim N\left(\beta_0+\beta_1x_0,\sigma^2h_{00}\right) \,\,\,\,\,\,\,\,\,\,\,\ \text{where} \,\,\,\,\,\,\,\,\,\,\,\ h_{00} = \frac{1}{n}+\frac{(x_0-\bar{x})^2}{(n-1)s_x^2} $$ with a confidence interval for $E(Y_0) =\beta_0+\beta_1x_0$ being $$ \left(\hat{b_0}+\hat{b_1}x_0-cs\sqrt{h_{00}},\hat{b_0}+\hat{b_1}x_0+cs\sqrt{h_{00}}\right) $$ where $c = t_{n-2,1-\frac{\alpha}{2}}$.

These results are found by looking at the shape of the distribution and at $E(\hat{Y_0})$ and $var(\hat{Y_0})$.

Predicting observations

We want to predict the observation $Y_0 = \beta_0+\beta_1x_0+\epsilon_0$ at a value $x_0$

$$ E(\hat{Y_0}-Y_0) = 0 \,\,\,\,\,\,\,\,\,\,\ \text{and}\,\,\,\,\,\,\,\,\,\ \text{Var}(\hat{Y_0}-Y_0) = \sigma^2(1+h_{00}). $$ Hence a prediction interval is of the form $$ \left(\hat{b_0}+\hat{b_1}x_0-cs\sqrt{h_{00}+1},\hat{b_0}+\hat{b_1}x_0+cs\sqrt{h_{00}+1}\right) $$

So I understand that in theory the prediction interval should be larger because of bigger uncertainty. I understand that we have this thanks to the $+1$ under square root.

BUT MY QUESTION IS: What is the difference in meaning between the two? What is the difference in mathematical terms between the two? For example, why on earth do we look at $E(\hat{Y_0}-Y_0)$ for the prediction interval? Of course we are going to get something different from before. Can you please give me an insight and intuition on why we do what we do and what we actually do?


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