Problem 1. Suppose y = Xβ + €, where y and e are vectors of length n, where β is a vector
of length k, and where X is n x (k+1) of rank k + 1 <n. Suppose that E[є] = 0, Suppose
the variance of the responses is constant and also each response is uncorrelated with any other
response. Use σ² to denote the variance of the responses. Finally, we suppose that the responses
are distributed as normal. Now we can write y ~ Νη (Χβ, σ21) and equivalently, ε ~ Nn(0, σ2Ι).
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(a) Suppose we conduct an F test for Ho: β₁ = 0 versus H1 31≠0 where β₁ =
(βι, β2,..., βκ). Write an expression for the F statistic in terms of SSR, SSE, σ², n, k and
any integers needed. What is the distribution of the F statistic when the null hypothesis
is true?
(b) Using the assumptions in part (a), write an expression for the F statistic that is equivalent
to the F statistic given in part (a); however use only R2, n, k and any integers needed.
(c) Given a random sample of 15 observations from a population that meets the stated as-
sumptions, where k = 2, ẞXTy = 900, yy = 950, and the average observed response is
7, find the F statistic and the R² statistic.
