Using the formulas on page 330 for \(\widehat{\alpha}\) and \(\widehat{\beta}\), show that

(a) the expression for \(\widehat{\alpha}\) is linear in the \(Y_{i}\)

(b) \(\widehat{\alpha}\) is an unbiased estimate of \(\alpha\)

(c) the expression for \(\widehat{\beta}\) is linear in the \(Y_{i}\)

(d) \(\widehat{\beta}\) is an unbiased estimate of \(\beta\)

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Least squares estimates Sxy ay-bx and B = Sxx where x and y are, respectively, the means of the values of x and y. The least squares estimates determine the best-fitting line Fitted (or estimated) regression line y = a + Bx where the on y, a, and indicates the estimated value. The individual deviations of the observations y; from their fitted values ya+x; are called the residuals. Residuals observation - fitted value = yi-a-Bxi