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I have a simple linear regression model $G: Y = \beta_0 + X\beta_1 + \epsilon$. I have found least square estimates for the coefficients, i.e. $\hat\beta_0 = 32.1359$ and $\hat\beta_1=-14.5388$. I have also found that the standard error for the estimate is $\hat \sigma$=2.72.

I have to find the standard error at a point $x=1.63$ and construct a 95% confidence interval for the expected value at that same point.

What I know is that $\hat Y \in N_n(X\beta, \sigma^2I_n)$, so $Y_{1.63}$ should also be normally distributed with expected value $\mu=(1, 1.63)\beta$ and variance $\sigma^2$.

How do I find such standard error and such confidence interval?

Thanks in advance!

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Assuming the errors are i.i.d $N(0,\sigma^2)$ with $\sigma$ unknown.

The mean response at $x=x_0$ is $$E(y\mid x_0)=\beta_0+\beta_1x_0$$

We estimate $E(y\mid x_0)$ from the fitted model by $$\hat\mu_{y\mid x_0}=\hat\beta_0+\hat\beta_1x_0$$

Show that

\begin{align} \operatorname{Var}(\hat\mu_{y\mid x_0})&=\operatorname{Var}(\hat\beta_0)+x_0^2\operatorname{Var}(\hat\beta_1)+2x_0\operatorname{Cov}(\hat\beta_0,\hat\beta_1) \\&=\sigma^2\left[\frac{1}{n}+\frac{(x_0-\bar x)^2}{S_{xx}}\right] \end{align}

, where $S_{xx}=\sum\limits_{i=1}^n (x_i-\bar x)^2$ as usual.

Since $\sigma$ is not specified, the estimated standard error of $\hat\mu_{y\mid x_0}$ is $$\text{S.E.}=\hat\sigma\sqrt{\frac{1}{n}+\frac{(x_0-\bar x)^2}{S_{xx}}}$$, where $\hat\sigma^2=\frac{1}{n-2}\sum\limits_{i=1}^n(y_i-\hat\beta_0-\hat\beta_1 x_i)^2$ is the residual variance.

Since $(\hat\beta_0,\hat\beta_1)$ is jointly normal, we have $$\frac{\hat\mu_{y\mid x_0}-E(y\mid x_0)}{\sigma\sqrt{\frac{1}{n}+\frac{(x_0-\bar x)^2}{S_{xx}}}}\sim N\left(0,1\right)$$

The above again is independent of $$\frac{(n-2)\hat\sigma^2}{\sigma^2}\sim \chi^2_{n-2}$$

So you have the pivot $$\frac{\hat\mu_{y\mid x_0}-E(y\mid x_0)}{\hat\sigma\sqrt{\frac{1}{n}+\frac{(x_0-\bar x)^2}{S_{xx}}}}\sim t_{n-2}$$

Hence a $100(1-\alpha)\%$ confidence interval for $E(y\mid x_0)$ has confidence limits $$\hat\mu_{y\mid x_0}\mp (t_{\alpha/2,n-2})\hat\sigma\sqrt{\frac{1}{n}+\frac{(x_0-\bar x)^2}{S_{xx}}}$$

Relevant threads (also check out the linked posts):

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