STAT 350

Assignment 3

1. The following table gives the carbon monoxide emission rates in grams per mile for two vehicles. Readings were taken approximately every 1000 miles for each car.

   VEHICLE 1 VEHICLE 2

   Mileage Emission Rate Mileage Emission Rate

   0 50 0 40

   1000 56 1100 49

   2000 58 2200 58

   3000 60 3000 65

   4200 58 4000 75

   5000 63 5300 77

   6000 73 6000 86

   6900 71 7000 93

   8000 76 8100 98

   9200 73 9000 103

   10000 80 10000 109

   1. Plot the data.
   2. Consider the following 4 models for the data:
      1. Two straight lines, one for each vehicle, with different slopes and intercepts,
      2. Two parallel straight lines.
      3. Two lines with the same intercept but different slopes.
      4. One straight line.

      Write out model equations for data points number 2 and 22 for the first 3 models. To be clear for the fourth model these equations would be

      

      and

      

      Other models would have different numbers of parameters of course.

   3. Fit all 4 models. Hand in: estimates of the slopes and intercepts and of . Do NOT just hand in output from SAS or Minitab.
   4. Using formal hypothesis tests and plots select the best of these models. Again, I do not want computer output but discussion. You may attach computer output in order to say things like: ``The Sum of Squares for ... is on page ..." and to hand in plots on which you comment in the discussion but I will not be looking through the output.
   5. For the final selected model estimate the total emissions of CO in grams for each vehicle over the first 10000 miles. (This is the area under the fitted straight line from 0 to 10000 and is a linear combination of the parameter estimates.) Attach a standard error.
   6. Are the emissions of the two vehicles different over the first 10000 miles?
   7. Suppose the two cars are of the same make but that one vehicle was equipped with a special pollution control device. In 4 or 5 sentences comment on the experimental design as a method of determining whether or not the device reduces emissions and on what else you would want to find out from the experimenter to help interpret the results.

      Some Help: Some of the models have design matrices which do not naturally have a column of ones. To fit these in SAS you will need to add / NOINT to the end of the model statement. So, for example, to fit a straight line relating emissions RATE and MILEAGE which passed through the origin you might use the statements

      proc glm
       model RATE = MILEAGE / NOINT ;

      You will also need to create one or more data files for SAS to read. The table above is available in the assignment lab. For some models you will have to create columns of the design matrix yourself, using some text editor such as Microsoft Word or whatever. You will have to use the model equations from part 2 to see what goes in the columns of the design matrix and then create a data set which has these columns in it.

      You may also want to use proc gplot to do the plots since these plots are much higher resolution. This procedure is just like proc plot but produces better graphs. Here is some SAS example code:

      data insure;
        infile 'insure.dat' firstobs=2;
        input year cost;
        code = year - 1975.5 ;
      proc glm  data=insure;
         model cost = code ;
         estimate 'fit1980' intercept 1 code  4.5  / E;
         output out=insfit p=fitted r=resid student=isr press=press rstudent=esr;
      run ;
      proc rank data=insfit out=qqdat normal=blom ;
         var resid;
         ranks nscores;
      run;
      proc gplot data=qqdat;
         plot resid*nscores;
      run;

      The output is here. The output statement produces internally standardized residuals is isr, press residuals in press and externally studentized residuals in esr. The purpose of proc rank is to compute the plotting points for a Q-Q plot and store the residuals together with the corresponding plotting points in a data set called qqdat. Then proc gplot plots them. You can use the on-line help for proc gplot to find out how to customize the axes.

      In MINITAB, too, it is possible to ask for no intercept in a regression model.

2. Data below are from a nitrogen balance experiment on Kangaroo Island Wallabies, taken from Barker,S. (1968). ``Nitrogen balance and Water Intake in the Kangaroo Island Wallaby'' Austral. J. Experimental Biology and Medical Science, 46, 17-32.

   Nitrogen Body Dry Water Nitrogen

   Excreted Weight Intake Intake Intake

   162 3.386 16.6 41.7 54

   174 3.033 18.1 40.9 99

   119 3.477 13.4 25.0 46

   205 3.278 22.6 39.2 188

   312 3.368 26.5 47.4 345

   157 2.932 21.4 51.6 66

   184 3.128 30.3 71.6 171

   155 3.251 17.6 27.1 81

   192 3.396 21.3 37.7 175

   331 3.497 29.9 50.5 399

   114 3.182 12.8 28.4 38

   159 3.234 19.6 34.3 106

   260 3.139 36.2 77.6 228

   265 3.434 35.0 58.9 291

   387 2.970 32.9 55.3 449

   146 3.230 22.9 46.2 72

   233 3.470 32.9 67.4 176

   261 3.000 35.7 77.1 235

   287 3.224 34.4 74.9 288

   412 3.366 36.2 60.7 485

   174 3.264 29.9 65.4 92

   171 3.292 21.7 51.2 126

   259 3.525 35.0 66.8 224

   298 3.036 29.7 65.8 276

   407 3.356 29.2 48.1 386

   Fit the model

   

   by least squares. Get estimates and standard errors for all the parameters and an estimate of . Suggest a simpler model for the data, and fit it. Check the fit of the model, graphically and, if the model seems poor, modify it appropriately. Hand in a dsicussion of your findings bolstered by output used only as an appendix. I will be marking the discussion, not sorting through the output.

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