In the problems below, you are given observational data \(\left\{\left(x_{i}, y_{i}ight)ight\}\), and information about \(\sigma\). Find the ...

- \(P \%=\left(1-\alpha_{1}ight) 100 \%\) credible interval \(I_{\alpha_{1}}\) for the regressions line \(y(x)\).

- \(Q \%=\left(1-\alpha_{2}ight) 100 \%\) predictive interval \(I_{\alpha_{2}}^{+}\)for the next observation \(Y_{+}(x)\).

(a) Data: \(\{(3,22),(7,16),(11,8),(4,21),(12,13),(17,3),(7,11),(6,14)\), \((13,10),(2,20),(15,9),(13,11),(14,14),(15,4),(8,8),(12,5),(16,3)\), \((6,9),(15,0),(2,19)\} . \sigma_{0}=4\) and \(n_{0}=7 . \alpha_{1}=0.08, \alpha_{2}=0.01\).

(b) Data: \(\{(10,16),(26,24),(22,20),(23,15),(26,28)\} . \sigma_{0}=0.5\) and \(n_{0}=\) 30. \(P \%=Q \%=93 \%\).

(c) Data: \(\{(10,16),(26,24),(22,20),(23,15),(26,28)\} . \sigma\) unknown. \(\alpha_{1}=\) \(\alpha_{2}=0.07\).