i'm trying to solve a problem that involve a linear model given its normal equations, and the errors have a normal distribution but i'm a little lost. the problem is about construct a 95% confidence interval to $\beta_{1},\beta_{2},\beta_{3}, \beta_{1}-\beta_{2}, \beta_{1}+\beta_{2}$

i was reading about confidence intervals and say that if you have a set of observations, and you have to create a confidence interval for a parameter $ \theta$ , then you need to construct a new random variable $h$ that satisfice a couple of properties:

  1. $h$ is a function of the observations and of the parameter
  2. the probability distribution of $ h$ , doesn't depend of $\theta$ or other parameters.

I'm so confused about how get that $h$ function. maybe i was thinking that

$h= h(Y_{1},Y_{2},...Y_{N},\beta)= \varepsilon$, specially cuz we now that errors have a normal distribution. But i'm really not sure, moreover i dont have the observations $Y_{i}$, just the normal equations. Can you explain me how to get that Confidence interval?.


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