Instructions: This is an open book test. You may use notes, text, other books and a calculator. Your presentations of statistical analysis will be marked for clarity of explanation. I expect you to explain what assumptions you are making and to comment if those assumptions seem unreasonable. The exam is out of 25.
You are asked for a CI for 25K. I tolerated prediction intervals for
a new Y, the measured work, for d=5 as well. The CI
is
Model # | Predictors |
1 | Abdomen Wrist Age Neck Biceps Thigh Chest |
Forearm Hip Ankle Height Weight Knee | |
2 | Abdomen Wrist Age Neck Biceps Chest Hip Height |
3 | Abdomen Wrist Height |
For Model I the overall F statistic is 51.34 with P=0.0001 so that any any reasonable level the hypothesis that all the coefficients are 0 is rejected. Thus the regression is indeed significant.
The extra SS F test is based on
The t-statistic for Age in model 2 is -1.34 with a P value of 0.1805. Thus Age is not significant; it can be dropped from the model.
The confidence interval is
In part b you concluded that model 2 improved on model 1 because it is simpler, eliminating insignificant terms. On the other hand you cannot eliminate all the terms you need to eliminate to get from model 2 to model 3 (either do another extra SS F test or just not that some of the deleted terms are quite significant according to the t tests in model 2). Thus Model 2 is to be preferred.
0.0099
The idea I was looking for was to try models which simplified Model 2 by eliminating some possibly insignificant terms. This would suggest eliminating HIP and AGE or perhaps HIP, AGE and NECK. I marked this one fairly liberally accepting for at least part marks quite a few explanations.
Dependent Variable: DENSITY Sum of Mean Source DF Squares Square F Value Pr > F Model 13 0.0655339 0.0050411 51.34 0.0001 Error 236 0.0231737 0.0000982 Corrected Total 249 0.0887076 R-Square C.V. Root MSE DENSITY Mean 0.738763 0.938523 0.0099 1.0558 Source DF Type I SS Mean Square F Value Pr > F ABDOMEN 1 0.0580635 0.0580635 591.31 0.0001 WRIST 1 0.0039993 0.0039993 40.73 0.0001 AGE 1 0.0010165 0.0010165 10.35 0.0015 NECK 1 0.0002624 0.0002624 2.67 0.1034 BICEPS 1 0.0003259 0.0003259 3.32 0.0698 THIGH 1 0.0000550 0.0000550 0.56 0.4549 CHEST 1 0.0003328 0.0003328 3.39 0.0669 FOREARM 1 0.0001583 0.0001583 1.61 0.2055 HIP 1 0.0006023 0.0006023 6.13 0.0140 ANKLE 1 0.0000732 0.0000732 0.75 0.3887 HEIGHT 1 0.0006151 0.0006151 6.26 0.0130 WEIGHT 1 0.0000280 0.0000280 0.29 0.5938 KNEE 1 0.0000015 0.0000015 0.02 0.9008 Source DF Type III SS Mean Square F Value Pr > F ABDOMEN 1 0.0101396 0.0101396 103.26 0.0001 WRIST 1 0.0012065 0.0012065 12.29 0.0005 AGE 1 0.0003498 0.0003498 3.56 0.0603 NECK 1 0.0002681 0.0002681 2.73 0.0998 BICEPS 1 0.0002004 0.0002004 2.04 0.1545 THIGH 1 0.0001884 0.0001884 1.92 0.1673 CHEST 1 0.0001730 0.0001730 1.76 0.1857 FOREARM 1 0.0001729 0.0001729 1.76 0.1858 HIP 1 0.0001719 0.0001719 1.75 0.1871 ANKLE 1 0.0001586 0.0001586 1.62 0.2050 HEIGHT 1 0.0001578 0.0001578 1.61 0.2062 WEIGHT 1 0.0000259 0.0000259 0.26 0.6081 KNEE 1 0.0000015 0.0000015 0.02 0.9008 T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 1.092703119 20.08 0.0001 0.05442893 ABDOMEN -0.002162705 -10.16 0.0001 0.00021283 WRIST 0.004350830 3.51 0.0005 0.00124125 AGE -0.000141394 -1.89 0.0603 0.00007492 NECK 0.000903669 1.65 0.0998 0.00054689 BICEPS -0.000566771 -1.43 0.1545 0.00039677 THIGH -0.000473593 -1.39 0.1673 0.00034192 CHEST 0.000334560 1.33 0.1857 0.00025207 FOREARM -0.000639418 -1.33 0.1858 0.00048183 HIP 0.000449416 1.32 0.1871 0.00033965 ANKLE -0.000650693 -1.27 0.2050 0.00051193 HEIGHT 0.000564160 1.27 0.2062 0.00044509 WEIGHT 0.000080268 0.51 0.6081 0.00015633 KNEE 0.000071344 0.12 0.9008 0.00057191
Dependent Variable: DENSITY Sum of Mean Source DF Squares Square F Value Pr > F Model 8 0.0649997 0.0081250 82.59 0.0001 Error 241 0.0237079 0.0000984 Corrected Total 249 0.0887076 R-Square C.V. Root MSE DENSITY Mean 0.732741 0.939380 0.0099 1.0558 Source DF Type I SS Mean Square F Value Pr > F ABDOMEN 1 0.0580635 0.0580635 590.24 0.0001 WRIST 1 0.0039993 0.0039993 40.65 0.0001 AGE 1 0.0010165 0.0010165 10.33 0.0015 NECK 1 0.0002624 0.0002624 2.67 0.1037 BICEPS 1 0.0003259 0.0003259 3.31 0.0700 CHEST 1 0.0003603 0.0003603 3.66 0.0568 HIP 1 0.0003324 0.0003324 3.38 0.0673 HEIGHT 1 0.0006394 0.0006394 6.50 0.0114 Source DF Type III SS Mean Square F Value Pr > F ABDOMEN 1 0.0116909 0.0116909 118.84 0.0001 WRIST 1 0.0010761 0.0010761 10.94 0.0011 AGE 1 0.0001774 0.0001774 1.80 0.1805 NECK 1 0.0002434 0.0002434 2.47 0.1170 BICEPS 1 0.0005632 0.0005632 5.72 0.0175 CHEST 1 0.0003868 0.0003868 3.93 0.0485 HIP 1 0.0001138 0.0001138 1.16 0.2831 HEIGHT 1 0.0006394 0.0006394 6.50 0.0114 T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 1.062471175 53.53 0.0001 0.01984842 ABDOMEN -0.002133222 -10.90 0.0001 0.00019568 WRIST 0.003778287 3.31 0.0011 0.00114235 AGE -0.000088111 -1.34 0.1805 0.00006561 NECK 0.000806964 1.57 0.1170 0.00051298 BICEPS -0.000841346 -2.39 0.0175 0.00035163 CHEST 0.000409888 1.98 0.0485 0.00020672 HIP 0.000280433 1.08 0.2831 0.00026068 HEIGHT 0.000748582 2.55 0.0114 0.00029363
Dependent Variable: DENSITY Sum of Mean Source DF Squares Square F Value Pr > F Model 3 0.0635560 0.0211853 207.21 0.0001 Error 246 0.0251516 0.0001022 Corrected Total 249 0.0887076 R-Square C.V. Root MSE DENSITY Mean 0.716467 0.957674 0.0101 1.0558 Source DF Type I SS Mean Square F Value Pr > F ABDOMEN 1 0.0580635 0.0580635 567.90 0.0001 WRIST 1 0.0039993 0.0039993 39.12 0.0001 HEIGHT 1 0.0014933 0.0014933 14.61 0.0002 Source DF Type III SS Mean Square F Value Pr > F ABDOMEN 1 0.0515675 0.0515675 504.37 0.0001 WRIST 1 0.0020169 0.0020169 19.73 0.0001 HEIGHT 1 0.0014933 0.0014933 14.61 0.0002 T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 1.071106097 58.02 0.0001 0.01845975 ABDOMEN -0.001770706 -22.46 0.0001 0.00007884 WRIST 0.004186145 4.44 0.0001 0.00094252 HEIGHT 0.001022304 3.82 0.0002 0.00026750