Extrapolation: The file swim.dat contains data on the amount of time, in seconds, it takes each of four high school swimmers to swim 50 yards. Each swimmer has six times, taken on a biweekly basis. a) Perform the following data analysis for each swimmer separately: i. Fit a linear regression model of swimming time as the response and week as the explanatory variable. To formulate your prior, use the information that competitive times for this age group generally range from 22 to 24 seconds. ii. For each swimmer j, obtain a posterior predictive distribution for Y j* , their time if they were to swim two weeks from the last recorded time. b) The coach of the team has to decide which of the four swimmers will compete in a swimming meet in two weeks. Using your predictive distributions, compute Pr(Y j = max{Y 1 *,...,Y 4* }|Y)) for each swimmer j, and based on this make a recommendation to the coach.

1st week 3rd week 5th week 7th week 9th week 11th week [1,] 23.1 23.2 22.9 22.9 22.8 22.7 [2,] 23.2 23.1 23.4 23.5 23.5 23.4 [3,] 22.7 22.6 22.8 22.8 22.9 22.8 [4,] 23.7 23.6 23.7 23.5 23.5 23.4