Statistics | Part IB, 2001

Consider the linear regression model

Yi=βxi+ϵi,Y_{i}=\beta x_{i}+\epsilon_{i},

i=1,,ni=1, \ldots, n, where x1,,xnx_{1}, \ldots, x_{n} are given constants, and ϵ1,,ϵn\epsilon_{1}, \ldots, \epsilon_{n} are independent, identically distributed N(0,σ2)N\left(0, \sigma^{2}\right), with σ2\sigma^{2} unknown.

Find the least squares estimator β^\widehat{\beta} of β\beta. State, without proof, the distribution of β^\widehat{\beta} and describe how you would test H0:β=β0H_{0}: \beta=\beta_{0} against H1:ββ0H_{1}: \beta \neq \beta_{0}, where β0\beta_{0} is given.

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