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:

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.

For each swimmer j, obtain a posterior predictive distribution

for Yj, their time if they were to swim two weeks from the last

recorded time.

swim.dat

23.1 23.2 22.9 22.9 22.8 22.7

23.2 23.1 23.4 23.5 23.5 23.4

22.7 22.6 22.8 22.8 22.9 22.8

23.7 23.6 23.7 23.5 23.5 23.4

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 dis- tributions,computePr(Yj =max{Y1,...,Y4}|Y))foreachswimmer j, and based on this make a recommendation to the coach.