Problem 1. A study was conducted on 97 men with prostate cancer who were due to receive a radical prostatectomy. The dataset prostate.dat contains data on 9 measurements made on these 97 men. (a) Fit a linear regression model with lpsa as the response and the other variables as the predictors (give the summary). Write down the equation to predict lpsa based on the other eight variables. Use standard noninformative priors. (b) Construct a 95% credible set for the coefficient of age. Can you conclude anything about its significance based on the credible set? (c) Predict lpsa of a 65-year old man with lcavol = 1.35, lweight = 3.65, lbph = 0.1, svi = .22, lcp = 0.18, gleason = 6.75, and pgg45 = 25 and construct its 95% credible set.

lcavol lweight agelbph svilcp gleason pgg45lpsa

-0.57981852.769550 -1.3862940 -1.3862960 -0.43078

-0.99425233.319658 -1.3862940 -1.3862960 -0.16252

-0.51082562.691274 -1.3862940 -1.38629720 -0.16252

-1.20397283.282858 -1.3862940 -1.3862960 -0.16252

0.75141613.432462 -1.3862940 -1.38629600.37156

-1.04982213.228850 -1.3862940 -1.38629600.76547

0.73716413.4735640.6151860 -1.38629600.76547

0.69314723.5395581.5368670 -1.38629600.85442

-0.77652883.539547 -1.3862940 -1.38629601.04732

0.22314363.244563 -1.3862940 -1.38629601.04732

0.25464223.604165 -1.3862940 -1.38629601.26695

-1.34707363.5987631.2669480 -1.38629601.26695

1.61342993.022963 -1.3862940 -0.597847301.26695

1.47704872.998267 -1.3862940 -1.38629751.34807

1.20597083.442057 -1.3862940 -0.43078751.39872

1.54115913.061166 -1.3862940 -1.38629601.44692

-0.41551543.5160701.2441550 -0.597847301.47018

2.28848623.649466 -1.38629400.37156601.49290

-0.56211893.267741 -1.3862940 -1.38629601.55814

0.18232163.8254701.6582280 -1.38629601.59939

1.14740253.419459 -1.3862940 -1.38629601.63900

2.05923883.5010601.47476301.348077201.65823

-0.54472723.375959 -0.7985080 -1.38629601.69562

1.78170913.4516630.43825501.178657601.71380

0.38526243.6674691.5993880 -1.38629601.73166

1.44691903.1246680.3001050 -1.38629601.76644

0.51282363.719765 -1.3862940 -0.798517701.80006

-0.40047763.8660671.8164520 -1.386297201.81645

1.04027673.1290670.22314400.048797801.84845

2.40964423.375965 -1.38629401.61939601.89462

0.28517894.0902651.9629080 -0.79851601.92425

0.18232166.1076651.7047480 -1.38629602.00821

1.27536283.0374711.2669480 -1.38629602.00821

0.00995033.267754 -1.3862940 -1.38629602.02155

-0.01005033.216963 -1.3862940 -0.79851602.04769

1.30833284.1198642.1713370 -1.38629752.08567

1.42310833.657173 -0.57981801.658238152.15756

0.45742482.374964 -1.3862940 -1.386297152.19165

2.66095864.0851681.37371611.832587352.21375

0.79750723.0131560.9360930 -0.16252752.27727

0.62057653.142060 -1.3862940 -1.386299802.29757

1.44220203.682668 -1.3862940 -1.386297102.30757

0.58221563.8660621.7137980 -0.43078602.32728

1.77155683.896961 -1.38629400.81093762.37491

1.48613973.4095661.7492000 -0.430787202.52172

1.66392613.3928610.6151860 -1.386297152.55334

2.72785283.9954791.87946512.6567691002.56879

1.16315084.0351681.7137980 -0.430787402.56879

1.74571553.498043 -1.3862940 -1.38629602.59152

1.22082993.5681701.3737160 -0.79851602.59152

1.09192333.993668 -1.3862940 -1.386297502.65676

1.66013104.2348642.0731720 -1.38629602.67759

0.51282363.6336641.49290400.048797702.68444

2.12704054.1215681.76644201.446927402.69124

3.15359043.516059 -1.3862940 -1.38629752.70471

1.26694764.2801662.1222620 -1.386297152.71800

0.97455962.865147 -1.38629400.50078742.78809

0.46373403.7647491.4231080 -1.38629602.79423

0.54232434.1782700.4382550 -1.386297202.80639

1.06125653.8512611.2947270 -1.386297402.81241

0.45742484.5245732.3263020 -1.38629602.84200

1.99741773.7197631.61938811.909547402.85359

2.77570883.524972 -1.38629401.558149952.85359

2.03470563.9170662.00821412.110217602.88200

2.07317193.623064 -1.3862940 -1.38629602.88200

1.45861503.8362611.3217560 -0.430787202.88759

2.02287123.8785681.78339101.321767702.92047

2.19833514.0509722.3075730 -0.430787102.96269

-0.44628714.408569 -1.3862940 -1.38629602.96269

1.19392254.7804722.3263020 -0.79851752.97298

1.86408013.593260 -1.38629411.321767603.01308

1.16002093.3411771.7492000 -1.386297253.03735

1.21491273.825469 -1.38629410.223147203.05636

1.83896113.2367600.43825511.178659903.07501

2.99922623.849169 -1.38629411.909547203.27526

3.14113053.263868 -0.05129312.420377503.33755

2.01089504.4338722.12226200.500787603.39283

2.53765724.3548782.3263020 -1.386297103.43560

2.64830023.582169 -1.38629412.584007703.45789

2.77944023.823263 -1.38629400.371567503.51304

1.46787433.0704660.55961600.223147403.51601

2.51365613.4735570.43825502.327287603.53076

2.61300673.888877 -0.52763310.559627303.56530

2.67759103.8384651.11514201.749209703.57094

1.56234633.7099601.69561600.810937303.58768

3.30284933.519064 -1.38629412.327287603.63099

2.02419313.7317581.6389970 -1.38629603.68009

1.73165553.369062 -1.38629410.300107303.71235

2.80759384.718165 -1.38629412.463857603.98434

1.56234633.6951760.93609310.810937753.99360

3.24649104.101868 -1.3862940 -1.38629604.02981

2.53290283.6776611.3480731 -1.386297154.12955

2.83026783.876468 -1.38629411.321767604.38515

3.82100363.896944 -1.38629412.169057404.68444

2.90744743.396252 -1.38629412.463857105.14312

2.88256363.7739681.55814511.558147805.47751

3.47196653.9750680.43825512.904177205.58293