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$$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|} \text{Obs}\# & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\\hline X & 5 & 8 & 10 & 4 & 5 & 12 & 2 & 6 & 3 & 6 \end{array}$$

These values are given and then we are provided with error terms, one for each observation, pulled from a normal distribution with mean=0 and variance=10 and we are supposed to generate the values of our dependent variable Y using the formula $Y_{i}=1+.5X_{i}+u_{i}$

The first question is asking: since we know the true properties of the data, then what is the true variance of $\hat{\beta_{o}}$ and $\hat{\beta_{1}}$. But I'm not really sure what that is asking: what is meant by true properties? Nothing was ever really explained or mentioned anything about true properties so I'm rather confused. Would anyone mind clarifying please?

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What is $\beta_0$ and $\beta_1$? – utdiscant Feb 8 '13 at 7:56
They are B0=1 and B1=.5 – USC Feb 8 '13 at 8:09

The true properties of the data are known, because you know exactly how it was generated. In more general problems of this type (well, also often, in real life...), you have no idea how the data is generated... Say that you're relating the price of gas (Y) with the price of crude oil on the market (x). It sounds like they would be (directly) related, but there are so many steps in between extracting oil and producing gas, that you do not know the formula that would let you compute Y directly from X.

In your case, it's not a real-world situation, but a study case. In other words, your life is made easier: you're given data that's generated in an ideal way, a completely specified way. You can take advantage of this!

As for computing the variances, lookup wikipedia for some good hints

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