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MA 2612 Test 4 D '98


A particular town employs three assessors who are responsible for determining the value of residential property. To see whether or not these assessors differ systematically in their assessments, five houses are selected and each assessor is asked to determine the market value of each house.

(10 points) Explain why this is not a completely random design.

ANS: It is a RCBD because all five houses are assessed by each assessor. The houses are blocks.

(10 points) Write out the model for this problem. Explain what each parameter means.

ANS: The model is the additive RCBD model

Y_{ij}=\mu+\tau_i+\gamma_j+\epsilon_{ij}, i=1,2,3,~j=1,\ldots,5,\end{displaymath}

where Yij is the assessed value given by assessor i to house j, $\mu$ is the overall population mean, $\tau_i$ is the effect due to assessor i, $\gamma_j$ is the effect of house j, and $\epsilon_{ij}$ is random error.

(10 points) The macro RCBD output the following information, as well as Figure 1. What do these tell you about the suitability of the model?

   F STATISTIC:    604E-7          DEGREES OF FREEDOM:     1    7
   P-VALUE:        0.994

ANS: The interaction plots show no evidence of interaction, since all line segments profiles are close to parallel. Tukey's test confirms this, as the large p-value shows.

(5 points) Figure 2 shows plots of studentized residuals versus Student's t quantiles and fitted values. Do these indicate a problem with the model? Give your reasons.

ANS: There is no evidence of a pattern in the plot of studentized residuals versus fitted values. The normal plot has some nonlinearity in both tails, which may indicate some nonnormality.

(10 points) The following is the ANOVA table for the fitted model. Are there significant differences in house valuations among assessors? Does it appear blocking was useful? Support your answers.

Analysis of Variance          
Source DF Sum of Squares Mean Square F Stat Prob > F
Assessor 2 356.1 178.1 63.22 0.0001
House 4 43852.7 10963.2 3892.25 0.0001
Error 8 22.5 2.8    
C Total 14 44231.3      

ANS: There appear to be significant differences among assessors (p-value: 0.0001). Blocking appears to be useful (p-value: 0.0001).

(15 points) Property taxes are based on assessments made by these assessors. If you own a house in the town, who do you want assessing your house, or doesn't it matter? The town manager lives in another town, so her taxes are not affected by which assessors are used. However, the town budget, including the budget of the town manager's office, is affected. Who does she want to hire as assessor, or doesn't it matter? Justify your answers based on the SAS output below.

                 General Linear Models Procedure

      Tukey's Studentized Range (HSD) Test for variable: Y

   NOTE: This test controls the type I experimentwise error rate.

       Alpha= 0.05  Confidence= 0.95  df= 8  MSE= 2.816667
           Critical Value of Studentized Range= 4.041
              Minimum Significant Difference= 3.033

Comparisons significant at the 0.05 level are indicated by '***'.

                    Simultaneous            Simultaneous
                        Lower    Difference     Upper
       ASSESSOR      Confidence    Between   Confidence
      Comparison        Limit       Means       Limit

     1    - 3          -2.033       1.000       4.033
     1    - 2           7.767      10.800      13.833   ***
     2    - 3         -12.833      -9.800      -6.767   ***
ANS: You want either 1 or 3, since they are essentially equivalent, statistically, and both assess significantly lower than 2, on average. The town manager wants assessments high, so she will want to hire assessor 2.

Figure 1:   Interaction plot, delivery data
\psfig {file=t2p2a_1.eps,height=2.5in,width=5in}

Figure 2:   Plots of studentized residuals versus Student's t quantiles (left) and fitted values (right)
\psfig {file=t2p2a_2.eps,height=2.5in,width=5in}

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Joseph D Petruccelli