So we have $100$ observations for $(x, y)$.
The mean of $x$ is $1.06$, and for $y$ it is $3$.
The standard deviation is $0.52$ for $x$ and for $y$ it is $1.13$.
the correlation between $x$ and $y$ is $0.89$.
In the question we are told to:
• Estimate the linear regression line of the regression of $Y$ on $X$ and the standard deviation of the errors.
• estimate the regression line when we regress $X$ as dependent variable on $Y$ and obtain an estimate of the standard deviation of the errors.
• Are the two regression lines the same? If not, then explain why not.
• For the regression of $Y$ on $X$, suppose that we wish to predict the
dependent variable $y$ at $x = x^* = 0.7$. Obtain the prediction, as
well as the standard error of the prediction.
• Obtain the standard deviation of the prediction error and hence
obtain a $95\%$ prediction interval for $y$ for the the given $x = x*.$
Now I thought we were supposed to generate $100$ points of data assuming $x$ and $y$ had a normal distribution with the given means and standard deviations, and then use stata to regress and find the prediction interval, etc
But I was told this was not the case by the lecturer, and was wondering if there was a way to solve this another way? I'm thinking some kind of derivation/calculations using the above info, but I have no idea where to start.
Best Answer
I've found estimates for $B_1$ and $B_0$ from modifying the formula used for their estimation; the numerator can be turned into $n \times cov(x,y)$; and we can find $cov(x,y)$ given $corr(x,y)$ and std of x and y.
Problem now is how to find the standard deviation of the errors and the prediction errors.