Part CCSIR NET December 2025centrality-of-the-chi-square-and-f-distributions-needs-the-null-but-independence-of-the-two-sums-of-squares-never-does

Centrality of the chi square and f distributions needs the null but independence of the two sums of squares never does

Let $Y=X\beta+\varepsilon$ be a multiple linear regression model with p regressors and an intercept, where $\beta=(\beta_{0},\beta_{1},\dots,\beta_{p})^{T}$ and the random error $\varepsilon$ ~ $N_{n}(0,\sigma^{2}I), \sigma>0$ and n>p+1. The least squares method provides a unique estimator $\beta$ ̂. Let the total sum of squares (corrected), sum of squares due to regression, and sum of squares due to error, based on $\beta$ ̂, be denoted YᵀAY, YᵀBY and YᵀCY respectively, so YᵀAY=YᵀBY+YᵀCY. Which of the following statements are always true?

A.YᵀAY $/\sigma^{2}$ follows a central $\chi^{2}$ distribution with (n−1) degrees of freedom.
B.YᵀBY $/\sigma^{2}$ follows a central $\chi^{2}$ distribution with p degrees of freedom if $\beta_{1}=\beta_{2}=\cdots=\beta_{p}=0$ .✓
C.YᵀBY/YᵀCY follows a central F distribution with (p,n−p) degrees of freedom.
D.YᵀBY and YᵀCY are independently distributed if and only if $\beta_{1}=\beta_{2}=\cdots=\beta_{p}=0$ .

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: OLS is the BLUE in every linear model

Centrality of the chi square and f distributions needs the null but independence of the two sums of squares never does

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