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Consider a simple linear regression model $Y = X\beta + \epsilon$ $ \ $. Let $Y_O$ be the least square predictor of $Y$ at $X = x_o$ , based on $n$ observations $ (X_i ,Y_i) $and $\overline X$ = sample mean for $ X_i $

then the standard error of the predictor $Y_O$

a) increases as $ x_o $ moves away from $\overline X$

b) decreases as $ x_o $ moves closer to 0

This is what I tried :- according to me the answer must be option a) because the standard error of the predictor is given by

$s\sqrt {\frac 1n + \frac {(x_o - \overline x )^2}{\sum {(x_i - \overline x)^2}}}$

so clearly if $ x_o $ moves away from $\overline X$ standard error increases, but the source where the answers are given (without explanation) says option 'b' is correct.

I am not able to understand how that is. I think it's a misprint, but still I wanted to confirm if from any approach option b) is more relevant than a)

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Your formula, reasoning and conclusion are right, so as you say, a is correct, b is (demonstrably) wrong.

Unfortunately there's not much to add; only if $\bar{x}$ were at $0$ would b be correct, but it would simply say the same thing as a in a different, less general way.

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