6.1 The Straight Line: 6.1.1 Confidence Intervals for the Slope and Intercept, 6.1.2 Confidence Interval for the X-Intercept, 6.1.3 Prediction Intervals and Bands, 6.1.4 Prediction Intervals for the Response, 6.1.5 Inverse Prediction (Calibration)

In fitting the straight line $Y_{i} = \beta_{0}+\beta_{1} x_{i}+\varepsilon_{i}(i = 1,2, \ldots , n)$, prove that $\bar{Y}$ and $\widehat{\beta }_{1}$ are uncorrelated. If $\widehat{Y}_{0} = \widehat{\beta }_{0}+\widehat{\beta }_{1} x_{0}$, deduce that

Using the notation of Section 6.1.2 , prove that $\widehat{\phi } = -\widehat{\beta }_{0} / \widehat{\beta }_{1}$ is the maximum likelihood estimate of $\phi$.

Given a general linear regression model, show how to find a confidence interval for the ratio $\mathbf{a}_{1}^{T} \mathbf{\beta } / \mathbf{a}_{2}^{T} \mathbf{\beta }$ of two linear parametric functions.

Using the notation of Section 6.1.5, show that when $\bar{x} = 0$,

where $r$ is the correlation coefficient of the pairs $\left(x_{i}, Y_{i}\right)$.