Let the independent random variables \(Y_{1}, \ldots, Y_{n}\) have the joint
pdf.
$$
L\left(\alpha, \beta, \sigma^{2}\right)=\left(\frac{1}{2 \pi
\sigma^{2}}\right)^{n / 2} \exp \left\\{-\frac{1}{2 \sigma^{2}}
\sum_{1}^{n}\left[y_{i}-\alpha-\beta\left(x_{i}-\bar{x}\right)\right]^{2}\right\\}
$$
where the given numbers \(x_{1}, x_{2}, \ldots, x_{n}\) are not all equal. Let
\(H_{0}: \beta=0(\alpha\) and \(\sigma^{2}\) unspecified). It is desired to use a
likelihood ratio test to test \(H_{0}\) against all possible alternatives. Find
\(\Lambda\) and see whether the test can be based on a familiar statistic. Hint:
In the notation of this section show that
$$
\sum_{1}^{n}\left(Y_{i}-\hat{\alpha}\right)^{2}=Q_{3}+\widehat{\beta}^{2}
\sum_{1}^{n}\left(x_{i}-\bar{x}\right)^{2}
$$