4.I 9H9 \mathrm{H} \quad

Statistics | Part IB, 2004

Suppose that Y1,,YnY_{1}, \ldots, Y_{n} are independent random variables, with YiY_{i} having the normal distribution with mean βxi\beta x_{i} and variance σ2\sigma^{2}; here β,σ2\beta, \sigma^{2} are unknown and x1,,xnx_{1}, \ldots, x_{n} are known constants.

Derive the least-squares estimate of β\beta.

Explain carefully how to test the hypothesis H0:β=0H_{0}: \beta=0 against H1:β0H_{1}: \beta \neq 0.

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