In an article in Statistics and Computing [An Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis (1991, pp. 119128)], Carlin and Gelfand investigated the age

In an article in Statistics and Computing [€œAn Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis€ (1991, pp. 119€“128)], Carlin and Gelfand investigated the age (x) and length (y) of 27 captured dugongs (sea cows).

(a) Find the least squares estimates of the slope and the intercept in the simple linear regression model. Find an estimate of σ².

(b) Estimate the mean length of dugongs at age 11.

(c) Obtain the fitted values y_i that correspond to each observed value y_i. Plot y_i versus y_i, and comment on what this plot would look like if the linear relationship between length and age were perfectly deterministic (no error). Does this plot indicate that age is a reasonable choice of regressor variable in this model?

3 1.0, 1.5, 1.5, 1.5, 2.5, 4.0, 5.0, 5.0, 7.0, 8.0, 8.5, 9.0, 9.5, 9.5, 10.0, 12.0, 12.0, 13.0, 13.0, 14.5, 15.5, 15.5, 16.5, 17.0, 22.5, 29.0, 31.5 1.80, 1.85, 1.87, 1.77, 2.02, 2.27, 2.15, 2.26, 2.47, 2.19, 2.26, 2.40, 2.39, 2.41, 2.50, 2.32, 2.32, 2.43, 2.47, 2.56, 2.65, 2.47, 2.64, 2.56, 2.70, 2.72, 2.57