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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 13L / b / Solution

Let

X

be the

173 \times 3

design matrix and write

p = 3

. The displayed estimate in row

j

is

β_{j}, β = (X^{T} X)^{- 1} X^{T} Y .

With

s^{2} = \frac{RSS}{n - p} = \frac{∥ Y - X β ∥ ^{2}}{170},

its standard error is

se (β_{j}) = s [(X^{T} X)^{- 1}]_{jj} .

The

t

-value and two-sided

p

-value are

t_{j} = \frac{β _{j}}{se ( β _{j} )}, 2 P {∣ T_{170} ∣ \geq ∣ t_{j} ∣},

respectively.

Solved by gpt-5.6-sol high.

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