A sample mean and a member of that sample share an observation. Do not assume they are independent merely because they are written with different symbols.
Rewrite before using variance rules
Let \(X_1,X_2,X_3\) be independent observations from \(N(\mu,\sigma^2)\), and let \(\overline{X}=\dfrac{X_1+X_2+X_3}{3}\). Then \[\overline{X}-X_1=\frac{-2X_1+X_2+X_3}{3}.\] Hence its mean is zero and its variance is \(\dfrac{(4+1+1)\sigma^2}{9}=\dfrac{2\sigma^2}{3}\). It is exactly normal.
Reasoning
Result
Incorrectly treat \(\overline{X}\) and \(X_1\) as independent
\(\dfrac{\sigma^2}{3}+\sigma^2=\dfrac{4\sigma^2}{3}\): invalid
Collect coefficients of the independent observations first
\(\overline{X}-X_1\sim N\left(0,\dfrac{2\sigma^2}{3}\right)\)
Bridge the idea
Now replace \(X_1\) by a new independent observation \(X_4\) from the same population. The samples no longer overlap, so \(\operatorname{Var}(\overline{X}-X_4)=\dfrac{4\sigma^2}{3}\). The different answer comes from the model, not a changed formula.
Exam wording: “The quantities share an observation, so independence cannot be assumed. Express the difference in terms of the original independent observations before calculating its variance.”

A sample mean and a member of that sample share an observation. Do not assume they are independent merely because they are written with different symbols.

Let \(X_1,X_2,X_3\) be independent observations from \(N(\mu,\sigma^2)\), and let \(\overline{X}=\dfrac{X_1+X_2+X_3}{3}\). Then \[\overline{X}-X_1=\frac{-2X_1+X_2+X_3}{3}.\] Hence its mean is zero and its variance is \(\dfrac{(4+1+1)\sigma^2}{9}=\dfrac{2\sigma^2}{3}\). It is exactly normal.

Incorrectly treat \(\overline{X}\) and \(X_1\) as independent

\(\dfrac{\sigma^2}{3}+\sigma^2=\dfrac{4\sigma^2}{3}\): invalid

Collect coefficients of the independent observations first

\(\overline{X}-X_1\sim N\left(0,\dfrac{2\sigma^2}{3}\right)\)

Now replace \(X_1\) by a new independent observation \(X_4\) from the same population. The samples no longer overlap, so \(\operatorname{Var}(\overline{X}-X_4)=\dfrac{4\sigma^2}{3}\). The different answer comes from the model, not a changed formula.

Exam wording: “The quantities share an observation, so independence cannot be assumed. Express the difference in terms of the original independent observations before calculating its variance.”

Shared Observations in Normal Averages

Shared Observations in Normal Averages