ࡱ> qk Nbjbj 4}},Gl...."dddd"(_aaaaaa$ ]yW..WWW.^_W_WW D@ pIvǂ"BdO&,0(u&WW""....Example of Model Diagnostics Calculator Maintenance Data Using EXCEL First, we begin with the original data. I have sorted it with respect to the predictor variable (X = number of machines serviced). Note that in this case we wish to preserve the pairs (Xi,Yi). To do this: Move the cursor into the field of data Click on Data on the main toolbar, then Sort Select Column 2 (X) and Ascending. If you have already placed headers on the columns, make sure you click on the correct option regarding headers. Y (minutes)X (Machines)101171332252393624534494785755655715685866977101710571188 Diagnostics for the Predictor Variable (Section 3.1) X-values that are far away from the rest of the others can exert a lot of influence on the least squares regression line. A histogram or bar chart of the X-values can identify any potential extreme values. The following steps in EXCEL can be used to obtain a histogram of the X-values. A copy of the histogram is given below the instructions. Select Tools on the header bar, then Data Analysis (you may need to add it in from add-ins), then Histogram For the Input Range, highlight the column containing X (if you have included the header cell, click on Labels). Click Chart Output then OK. You may experiment and make the chart more visually appealing if preparing reports, but for investigating the model assumptions, this is fine.  Residuals (Section 3.2) The model assumptions are that the error terms are independent and normally distributed with mean 0 and constant (with respect to levels of X) variance s2. The errors are:  EMBED Equation.3  Since the model parameters are unknown, we cannot observe the actual errors. However, if we replace the unknown parameters, we have an estimate of each residual by taking the difference between the actual and fitted values. These are referred to as the residuals:  EMBED Equation.3  These residuals should approximately demonstrate the same behavior as the true error terms (the approximation will be better as the sample size increases). Some important properties concerning the residuals: Mean:  EMBED Equation.3  Shown in Chapter 1. Thus, the residuals have mean 0 Variance:  EMBED Equation.3  Independence: Residuals are not independent due to:  EMBED Equation.3  For large samples, relative to the number of model parameters, the dependency is unimportant. Note that under the model assumptions, if we standardize the errors by subtraction off their mean (which is 0) and divide through by their standard deviation, then they have a standard normal (Z) distribution:  EMBED Equation.3  Semistudentized Residuals are quantities that approximate the standardized errors, based on the fitted equation. They are based on the estimates of the unknown errors (the residuals) and the estimate of the error standard deviation. These can be used to identify outlying observations since these are like Z-scores:  EMBED Equation.3  Note that the residuals have complicated standard deviations that are not constant (we will pursue this later in course), so this is an approximation. EXCEL produces Standardized Residuals, which appear to be computed as:  EMBED Equation.3  The denominator is the square root of the average variance of the residuals. Note as the sample size increases these are very similar quantities. For purposes of identifying outlying observations, either of these is useful. Obtaining Residuals in EXCEL Choose Tools, Data Analysis, Regression Highlight the column containing Y, then the column containing X, then the appropriate Labels option Click on Residuals and Standardized Residuals Click OK The residuals will appear on a worksheet below the ANOVA table and parameter estimates. Also printed are observation number, predicted (fitted) values, and standardized residuals. Regression StatisticsMultiple R0.990215218R Square0.980526177Adjusted R Square0.979309063Standard Error4.481879999Observations18 ANOVAdfSSMSFSignificance FRegression116182.616182.68064.09733E-15Residual16321.420.1Total1716504 ObservationPredicted Y (minutes)ResidualsStandard Residuals112.41610738-2.416107383-0.555674513212.416107384.5838926171.054238034327.154362425.8456375841.344423613427.15436242-2.154362416-0.495476441541.89261745-2.89261745-0.665265875656.630872485.3691275171.234832251756.63087248-3.630872483-0.83505531856.63087248-7.630872483-1.755005337971.369127526.6308724831.525017831071.369127523.6308724830.835055311171.36912752-6.369127517-1.4648197581271.36912752-0.369127517-0.0848947171371.36912752-3.369127517-0.7748572371486.10738255-0.10738255-0.02469664515100.8456376-3.845637584-0.884448616100.84563760.1543624160.03550142717100.84563764.1543624160.95545145418115.58389262.4161073830.555674513 Diagnostics for Residuals (3.3) Obtaining a Plot of Residuals Against X (ei vs Xi) Copy and paste the column of Residuals to the original spreadsheet in Column C. Highlight Columns B and C and click on the Chart Wizard icon Click on XY (Scatter) then click through the dialog boxes Using all default options, your plot will appear as below. Y (minutes)X (Machines)Residuals101-2.416111714.5838933325.845638252-2.15436393-2.892626245.369128534-3.63087494-7.630877856.6308727553.630872655-6.36913715-0.36913685-3.36913866-0.10738977-3.8456410170.15436210574.15436211882.416107  Plots of residuals versus predicted values and residuals versus time order (when data are collected over time) would be obtained in similar manners. Simply copy and paste columns of interest to new columns, placing the variable to go on the horizontal (X) axis to the left of the variable to go on the vertical (Y) axis. Normality of Errors The simplest way to check for normality of the error terms is to obtain a histogram of the residuals. There are several ways to do this, the simplest being as follows: Choose Tools, Data Analysis, Histogram Highlight the column containing the Residuals Choose appropriate Labels choice Click Chart Output then OK A crude histogram will appear which is fine for our purposes. You may wish to experiment with EXCEL to obtain more elegant plots.  Note that you can choose bin upper values that are more satisfactory. Type in desired upper endpoints of bins in a new range of cells Choose Tools, Data Analysis, Histogram Highlight the column containing the Residuals For Bin Range highlight the range of values youve entered (include a label) Choose appropriate Labels choice Click on Chart Output then OK residual-7.5-2.52.57.5 The ranges will be:  EMBED Equation.3   Computing Expected Residuals Under Normality Copy the cells containing Observation and Residuals to a new worksheet in Columns A and B, respectively. Highlight the column of Residuals then select Data and Sort then click on Continue with Current Selection then OK. Note that the residuals are in ascending order and the observation number represents the rank now, as opposed to i Compute the percentile representing each residual in their empirical distribution. Go to Cell C2 (assuming that you have a header row with labels). Then type: =((A2-0.375)/(n+0.25)) where n is the sample size (type the number) Highlight Cell C2, then Copy it. Then highlight the next n-1 cells in column C, then Paste. Compute the Z values from the standard normal distribution corresponding to the percentiles in column C. Go to Cell D2 (assuming that you have a header row with labels). Then type: =NORMSINV(C2) Highlight Cell D2, then Copy it. Then highlight the next n-1 cells in column D, then Paste. Compute the Expected residuals under normality by multiplying the elements of Column D by  EMBED Equation.3  . This could be done in Column E. The results of the steps are shown below: First, put observation number and residuals in a new worksheet: ObservationResiduals1-2.4161124.58389335.8456384-2.154365-2.8926265.3691287-3.630878-7.6308796.630872103.63087211-6.3691312-0.3691313-3.3691314-0.1073815-3.84564160.154362174.154362182.416107 Second, sort only the residuals: ObservationResiduals1-7.630872-6.369133-3.845644-3.630875-3.369136-2.892627-2.416118-2.154369-0.3691310-0.10738110.154362122.416107133.630872144.154362154.583893165.369128175.845638186.630872 Third, compute the percentiles (notice that they are symmetric around 0.5). Here n=18 ObservationResidualspercentile1-7.630870.0342472-6.369130.0890413-3.845640.1438364-3.630870.198635-3.369130.2534256-2.892620.3082197-2.416110.3630148-2.154360.4178089-0.369130.47260310-0.107380.527397110.1543620.582192122.4161070.636986133.6308720.691781144.1543620.746575154.5838930.80137165.3691280.856164175.8456380.910959186.6308720.965753 Fourth, compute the Z-values from the standard normal distribution corresponding to the percentiles for the ordered residuals:  EMBED Equation.3  ObservationResidualspercentilez(pct)1-7.630870.034247-1.821752-6.369130.089041-1.346683-3.845640.143836-1.063244-3.630870.19863-0.846525-3.369130.253425-0.663756-2.892620.308219-0.50097-2.416110.363014-0.350418-2.154360.417808-0.20759-0.369130.472603-0.0687310-0.107380.5273970.068728110.1543620.5821920.207503122.4161070.6369860.350415133.6308720.6917810.500904144.1543620.7465750.663752154.5838930.801370.846524165.3691280.8561641.063245175.8456380.9109591.346684186.6308720.9657531.821745 Fifth, multiply the residual standard error ( EMBED Equation.3 ) by the Z-values to obtain the expected residuals under normality. ObservationResidualspercentilez(pct)expected 1-7.630870.034247-1.82175-8.161422-6.369130.089041-1.34668-6.033153-3.845640.143836-1.06324-4.763344-3.630870.19863-0.84652-3.792435-3.369130.253425-0.66375-2.973616-2.892620.308219-0.5009-2.244057-2.416110.363014-0.35041-1.569868-2.154360.417808-0.2075-0.929629-0.369130.472603-0.06873-0.307910-0.107380.5273970.0687280.307903110.1543620.5821920.2075030.929616122.4161070.6369860.3504151.569858133.6308720.6917810.5009042.244051144.1543620.7465750.6637522.973607154.5838930.801370.8465243.792426165.3691280.8561641.0632454.763337175.8456380.9109591.3466846.033145186.6308720.9657531.8217458.161418 Obtaining a Normal Probability Plot Copy the Residuals column to the right-hand side of the Expecteds column Highlight these 2 columns Click on Chart Wizard, then XY (Scatter), then click thru dialog boxes ObservationResidualspercentilez(pct)expected Residuals1-7.630870.034247-1.82175-8.16142-7.630872-6.369130.089041-1.34668-6.03315-6.369133-3.845640.143836-1.06324-4.76334-3.845644-3.630870.19863-0.84652-3.79243-3.630875-3.369130.253425-0.66375-2.97361-3.369136-2.892620.308219-0.5009-2.24405-2.892627-2.416110.363014-0.35041-1.56986-2.416118-2.154360.417808-0.2075-0.92962-2.154369-0.369130.472603-0.06873-0.3079-0.3691310-0.107380.5273970.0687280.307903-0.10738110.1543620.5821920.2075030.9296160.154362122.4161070.6369860.3504151.5698582.416107133.6308720.6917810.5009042.2440513.630872144.1543620.7465750.6637522.9736074.154362154.5838930.801370.8465243.7924264.583893165.3691280.8561641.0632454.7633375.369128175.8456380.9109591.3466846.0331455.845638186.6308720.9657531.8217458.1614186.630872  As always, you can make the plot more attractive with plot options, but it is unnecessary for our purposes of assessing normality. For this example, the residuals appear to fall on a reasonably straight line, as would be expected under the normality of errors assumption. Correlation Test for Normality (3.5)  EMBED Equation.3 Error terms are normally distributed  EMBED Equation.3 Error terms are not normally distributed TS: Correlation coefficient between observed and expected residuals ( EMBED Equation.3 ) RR:  EMBED Equation.3  Tabled values in Table B.6, Page 1348 (indexed by a and n) We can obtain the correlation coefficient between the observed and expected residuals as follows. Select Tools, Data Analysis, Correlation Highlight the columns for Residuals and Expected Click on Labels if they are included Click OK expected Residualsexpected 1Residuals0.9808161 For this example, n=15 and with  EMBED Equation.3 , we obtain a critical value of 0.946. Since the correlation coefficient (0.981) is larger than the critical value, we conclude in favor of the null hypothesis. We conclude that the errors are normally distributed. Modified Levene Test for Constant Variance (3.6) To conduct this test in EXCEL, do the following steps: Split the data into two groups with respect to levels of X. Use best judgment in terms of balance and closeness of X levels. For our example a natural split is group 1: X = 1-4 and group 2: X = 5-8 Obtain the Residuals from the regression. In a new worksheet put the residuals from group 1 in one column (say Column A), the residuals from group 2 in another column (say Column B). For this example, the group sizes are  EMBED Equation.3  Obtain the Median residual for each group. In Cell A15, type: =median( A2:A9) (since we have n1=8 and a header row). In Cell B15, type: =median( B2:B11) (since we have n2=10 and a header row). Obtain the absolute values of the differences between the residuals and their group medians in the next two columns. In Cell C2 type: =abs(A2-$A$15) (the dollar signs make cut and paste work correctly) Then Copy Cell C2 and Paste it to Cells C3-C9 In Cell D2 type: =abs(B2-$B$15) Then Copy Cell D2 and Paste it to Cells D3-D11 Obtain the mean and sum of squared deviations of the absolute difference from the median in the previous step. In Cell F2 type: =average(C2:C9) (this computes  EMBED Equation.3 ) In Cell F3 type: =devsq(C2:C9) (this computes  EMBED Equation.3 ) In Cell G2 type: =average(D2:D11) (this computes  EMBED Equation.3 ) In Cell G3 type: =devsq(D2:D11) (this computes  EMBED Equation.3 ) Compute s2 . In Cell H2 type: =(F3+G3)/(18-2) (18=n) Compute  EMBED Equation.3 . In Cell I2 type: =(F2-G2)/sqrt(H2*((1/8)+(1/10))) (since n1=8 and n2=10) The result of the steps on the calculator maintenance are shown below. First, separate the residuals into Columns A and B: Group 1Group 2-2.416116.6308724.5838933.6308725.845638-6.36913-2.15436-0.36913-2.89262-3.369135.369128-0.10738-3.63087-3.84564-7.630870.1543624.1543622.416107 Second, obtain the median residuals for each group: Group 1Group 2-2.416116.6308724.5838933.6308725.845638-6.36913-2.15436-0.36913-2.89262-3.369135.369128-0.10738-3.63087-3.84564-7.630870.1543624.1543622.416107-2.285230.02349 Third, obtain the absolute difference between the actual residuals and the group medians: Group 1Group 2d1d2-2.416116.6308720.1308726.6073834.5838933.6308726.8691283.6073835.845638-6.369138.1308726.392617-2.15436-0.369130.1308720.392617-2.89262-3.369130.6073833.3926175.369128-0.107387.6543620.130872-3.63087-3.845641.3456383.869128-7.630870.1543625.3456380.1308724.1543624.1308722.4161072.392617-2.285230.02349 Fourth, compute the statistics: mean and sum of squared deviations for the d values for groups 1 and 2: Group 1Group 2d1d2stats 1stats 2-2.416116.6308720.1308726.6073833.7768463.1046984.5838933.6308726.8691283.60738388.5585150.601915.845638-6.369138.1308726.392617-2.15436-0.369130.1308720.392617-2.89262-3.369130.6073833.3926175.369128-0.107387.6543620.130872-3.63087-3.845641.3456383.869128-7.630870.1543625.3456380.1308724.1543624.1308722.4161072.392617-2.285230.02349 Fifth, Compute the pooled variance s2: Group 1Group 2d1d2stats 1stats 2pooled s^2-2.416116.6308720.1308726.6073833.7768463.1046988.6975264.5838933.6308726.8691283.60738388.5585150.601915.845638-6.369138.1308726.392617-2.15436-0.369130.1308720.392617-2.89262-3.369130.6073833.3926175.369128-0.107387.6543620.130872-3.63087-3.845641.3456383.869128-7.630870.1543625.3456380.1308724.1543624.1308722.4161072.392617-2.285230.02349 Sixth, compute the test statistic  EMBED Equation.3  Group 1Group 2d1d2stats 1stats 2pooled s^2t-stat-2.416116.6308720.1308726.6073833.7768463.1046988.6975260.480484.5838933.6308726.8691283.60738388.5585150.601915.845638-6.369138.1308726.392617-2.15436-0.369130.1308720.392617-2.89262-3.369130.6073833.3926175.369128-0.107387.6543620.130872-3.63087-3.845641.3456383.869128-7.630870.1543625.3456380.1308724.1543624.1308722.4161072.392617-2.285230.02349 Finally, we can conduct the test: For a = 0.05, we obtain:  EMBED Equation.3 . Since our test statistic (0.48) does not exceed 2.120, we fail to reject the hypothesis of equal variances. We have no reason to believe that the error variance is not constant.  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  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~6Root Entry Fvǂ?Data WWordDocument4ObjectPool'0%tǂvǂ_1125388498Fp(tǂ.tǂOle CompObjfObjInfo  #$'*+,/2369:=@ABEJMPUZ_dgjoruz} FMicrosoft Equation 3.0 DS Equation Equation.39ql]  i =Y i "E{Y i }=Y i "( 0 + 1 X i )Equation Native _1125388832E FU3tǂp4tǂOle CompObj f FMicrosoft Equation 3.0 DS Equation Equation.39ql] e i =Y i "'Y ^  i =Y i "(b 0 +b 1 X i )ObjInfo Equation Native  _1125389264F7tǂPp9tǂOle  FMicrosoft Equation 3.0 DS Equation Equation.39qRl] e i =0!e  " =0 FMicrosoft Equation 3.0 DS EqCompObjfObjInfoEquation Native n_1125389402 F:tǂ0>tǂOle CompObjfObjInfoEquation Native uation Equation.39ql] s 2 =MSE=SSEn"2=(e i "e) 2 " n"2 FMicrosoft Equation 3.0 DS Eq_1125389610FpAtǂBtǂOle CompObjfObjInfo!uation Equation.39qil] e i =X i e i =0  "  " FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native "_1125389963"FPEtǂHtǂOle %CompObj &fObjInfo!(Equation Native )_1125390618$FпKtǂpFMtǂOle -l]  i* = i "E{ i }= i "0= i ~N(0,1) FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj#%.fObjInfo&0Equation Native 1_1125390937;)FSPtǂPQtǂwl] e i* =e i s=e i  MSE  FMicrosoft Equation 3.0 DS Equation Equation.39qOle 4CompObj(*5fObjInfo+7Equation Native 8l] e i** =e i  MSE1"1n[]  FMicrosoft Equation 3.0 DS Equation Equation.39ql] ("","7.      !#"$CE&'()*+,-./0123456789:;<=>?@ABDnFpHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmostuvwxyz{|}~_1125398029.F7gtǂQmtǂOle ;CompObj-/<fObjInfo0>Equation Native ?_112540057563FlstǂPyvtǂOle CCompObj24Df5]("7.5,"2.5]("2.5,2.5](2.5,7.5](7.5,") FMicrosoft Equation 3.0 DS Equation Equation.39ql]  MSE ObjInfo5FEquation Native G:_11254019518FA~tǂ tǂOle H FMicrosoft Equation 3.0 DS Equation Equation.39q5l] P(Zd"z(A))=A FMicrosoft Equation 3.0 DS EqCompObj79IfObjInfo:KEquation Native LQ_11254024531^=Ftǂ@itǂOle NCompObj<>OfObjInfo?QEquation Native R:uation Equation.39ql]  MSE  FMicrosoft Equation 3.0 DS Equation Equation.39ql] H 0 :_1123519592BFtǂtǂOle SCompObjACTfObjInfoDVEquation Native W;_1123519701@GF$tǂtǂOle XCompObjFHYf FMicrosoft Equation 3.0 DS Equation Equation.39ql] H A : FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoI[Equation Native \;_1125403739LFƜtǂLtǂOle ]CompObjKM^fObjInfoN`Equation Native a>_1123519846QFtǂtǂ"l] r ee* FMicrosoft Equation 3.0 DS Equation Equation.39q'l] r ee* d"Ole bCompObjPRcfObjInfoSeEquation Native fC_1125403842JYVF@tǂtǂOle hCompObjUWifObjInfoXk FMicrosoft Equation 3.0 DS Equation Equation.39q!l] =0.05 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native l=_1125405599[F"tǂ`tǂOle mCompObjZ\nfObjInfo]pEquation Native qa_1125415623Th`F@=tǂùtǂOle sEl] n 1 =8n 2 =10 FMicrosoft Equation 3.0 DS Equation Equation.39q!l] d 1CompObj_atfObjInfobvEquation Native w=_1125415794eFtǂtǂOle xCompObjdfyfObjInfog{Equation Native |{ FMicrosoft Equation 3.0 DS Equation Equation.39q_l] (d i1 "d 1 ) 2 " FMicrosoft Equation 3.0 DS Eq_1125415901crjFtǂ tǂOle ~CompObjikfObjInfoluation Equation.39q!l] d 2 FMicrosoft Equation 3.0 DS Equation Equation.39q_yd (d i2Equation Native =_1125415918oFtǂUtǂOle CompObjnpfObjInfoqEquation Native {_1125416308mwtFtǂ0tǂOle  "d 2 ) 2 " FMicrosoft Equation 3.0 DS Equation Equation.39ql] t L* FMicrosoft Equation 3.0 DS EqCompObjsufObjInfovEquation Native ;_1125418390|yFtǂP8tǂOle CompObjxzfObjInfo{Equation Native ;uation Equation.39ql] t L* FMicrosoft Equation 3.0 DS Equation Equation.39ql] t(1"(/_1125418658~FRtǂptǂOle CompObj}fObjInfoEquation Native _1125254823O FPmtǂptǂOle CompObjb2);n"2)=t(0.975;16)=2.120 FMicrosoft Excel ChartBiff8Excel.Sheet.89qOh+'08@Xp  Larry WiObjInfoWorkbook%";SummaryInformation(DocumentSummaryInformation8  Ba= f =K|8X1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1'Arial1'Arial1cArial1'Arial1'Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)                + ) , *   p  p "x@  Chart1Sheet2Sheet1O!Sheet35+Sheet45calcmainjb( 3  @@  > Y (minutes) X (Machines)BinMore FrequencyF nwnVw5n 0dT03ndowbT0n<3ndwbT0<n  Ir04t4tM MM`8tS0t5n3n3n*n5nǖwn5n3nw5ndw5nn5n3n3n*n5nǖwn5n3nw5n$dw5nOwn6n???bT03n owbT0n3nwbT04;nwwn+w,oww\wZwjT0bT0n;nn8vD0jT0P`jT0Lh0bT0bT00K0bT0jT0Ls0j0j8v 0j  "??3` o ` o ` o ` o =3d23 M NM4 3Q  FrequencyQ ;Q ;Q3_4E4D $% M 3O&Q4$% M 3O&Q4FAW? 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M43*43" 44% FRM03OU&Q  Histogram'44eee   >@    dMbP?_*+%MHP LaserJet 4000 PSW odXLetterPRIV''''b"dX??U} } m  X (Machines)T0 v a v v vvvv~ Y (minutes) X (Machines)$@?1@?@@@9@@C@@O@@J@@H@@ S@@ R@@ @P@@ Q@@ Q@@U@@@X@@@Y@@@Z@@]@ @*9h1>@ nnerx Larry WinnerxMicrosoft Excel@Q|՜.+,D՜.+,d  PXp x UF Statisticsma  Sheet2Sheet1Sheet3Sheet4 calcmainChart1  WorksheetsCharts 6> _PID_GUIDAN{85BE8B75-C006-466D-A13C-BC8011E06B7A} FMicrosoft Excel ChartBiff8Excel.Sheet.89qOh+'08@Xp  Larry Wi_1125393384 FP uǂuǂOle CompObjbObjInfoWorkbookGLMSummaryInformation(DocumentSummaryInformation8_1125395031, Fp!uǂ@[uǂ  Ba= f =Kj8X1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1Arial1'Arial1'Arial1cArial1'Arial1'Arial1Arial1Arial1Arial1Arial"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)                + ) , *   p  p "x@ &x@  Chart1wSheet2Sheet1-#Sheet3m-Sheet4?calcmainjb( 3  @@  +" Y (minutes) X (Machines)BinMore FrequencySUMMARY OUTPUTRegression Statistics Multiple RR SquareAdjusted R SquareStandard Error ObservationsANOVA RegressionResidualTotal InterceptdfSSMSFSignificance F Coefficientst StatP-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%RESIDUAL OUTPUT ObservationPredicted Y (minutes) ResidualsStandard Residuals" 4x'Y  6EtMMN 0Ir0.Et.EtM MMHEtS0t00MMM^n00 00Ԣ00HEttb0DtPercent0]y4>tp$Ν0\l0ZT0_T0=l0rTe0Tr0rTdrr? 0dr)@NJT0e0YT0L0@,|0v<v~0BT0Lv`vu0(1T0|0vj0jv 0jE 000  "pip Squ?? 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Y (minutes) X (Machines)BinMore FrequencySUMMARY OUTPUTRegression Statistics Multiple RR SquareAdjusted R SquareStandard Error ObservationsANOVA RegressionResidualTotal InterceptdfSSMSFSignificance F Coefficientst StatP-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%RESIDUAL OUTPUT ObservationPredicted Y (minutes) ResidualsStandard Residuals" ( x  'M @,(< w0Vw25 cT03ow*bT03w*bT0X .EtM >XwXwptowrwt80 Hw;woww_@>w_@538q*5ǖw53w5 dw5`5334>tp,Ν05l5tdw5OwcX6???bT03 owcbT0l3w@,|0v<v~0BT0Lv`vu0(1T0|0vr0rv 0rE 000  "pip Squ?? S3` o ` o ` o ` o _п3d23 M NM4 3Q  FrequencyQ ;Q ;Q3_4E4D $% M 3O&Q4$% M 3O&Q4FA{Bu3OfV ] 3 b43*4% z M3O &Q  Bin'4% `MZ3OT&Q  Frequency'4523  O43"  _aJ3O _% M3OQ4444A{Bu3O3 b! 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M43" 44%  MM03O~+&Q  Histogram'44eee >@    dMbP?_*+%MHP LaserJet 4000 PSW odXLetterPRIV''''b"dX??U} } m}  } $}  *  T0            v t  "     w    g   T  ' wG G wT SUMMARY OUTPUTRegression Statistics Multiple RPׯ?R Square-knx`?Adjusted R SquareqV?Standard Error6Jq@ Observations~ 2@ ANOVA   df SS MS F Significance F   Regression~ ? mPM@ mPM@ ,@ Yms<  Residual~ 0@ JAUt@ JAU4@  Total 1@@  CoefficientsStandard Errort StatP-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Interceptt%“9d̃@0@F#;?`7@`7@ X (Machines)DxRy-@x,?w9$b<@yms<ecF+@O/@ecF+@O/@RESIDUAL OUTPUT ObservationPredicted Y (minutes) ResidualsStandard Residuals~ ?[ (@Pl}0TnЈ῾~ @[ (@IAU@u(?~ @K';@PYa@7TU‚?~ @K';@`M\"</߿~ @IAD@p.$8%I忾~ @mPL@@xRy@d _v?~ @mPL@[ /Ÿ꿾~  @mPL@o2y<| 5<A?HE?,p}[>6XXXXXXX T0! 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M4523  O43d"  ,A3O ,% M3OQ443_ M MM  MM<444% L_MЙ3O=Q'44eee >@ SummaryInformation(DocumentSummaryInformation8_1125398574 FkuǂTzuǂOle nnerx Larry WinnerxMicrosoft Excel@Q|՜.+,D՜.+,p, PXp x UF Statisticsma  Sheet2Sheet1Sheet3Sheet4Sheet6 calcmainChart2  WorksheetsCharts 6> _PID_GUIDAN{7E03CEF1-D25A-4E4A-9A16-084CDAA7130B} FMicrosoft Excel ChartBiff8Excel.Sheet.89qOh+'08CompObjbObjInfoWorkbookK{SummaryInformation(  Ba=  f =KR8X1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1'>Arial1'>Arial1c>Arial1'>Arial1'>Arial1>Arial1>Arial1>Arial1>Arial11>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial1>Arial"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)                + ) , *   p  p "x@ &x@  t Chart1iSheet2w"Sheet1&Sheet3_0Sheet4~BSheet6LSheet7VSheet5bSheet8tlcalcmain , H3  @@  9% Y (minutes) X (Machines)BinMore FrequencySUMMARY OUTPUTRegression Statistics Multiple RR SquareAdjusted R SquareStandard Error ObservationsANOVA RegressionResidualTotal InterceptdfSSMSFSignificance F Coefficientst StatP-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%RESIDUAL OUTPUT ObservationPredicted Y (minutes) ResidualsStandard Residuals-7.5residual " h x  ' Vw;5 cT0 Xa3bT0D3lw30DXQ Ir0XaXaM, MMYaS0t00533*^n00 00Ԣ00Yatb0XaPercent0]w5,dw5Qap Ν0?lbT03 owtbT0lQ3wbT0<;ww`T+w4owwE w@,|0v<v~0BT0Lv`vu0(1T0|0v0v 0E 0<#00#  "pip Squ?? 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M43" 44%  MM03O~+&Q  Histogram'44eee >@    dMbP?_*+%MHP LaserJet 4000 PSW odXLetterPRIV''''b"dX??U} } m}  } $}  *  T0            v t v v!   v  v  vv ~    v  & v ~ v |0SUMMARY OUTPUTRegression Statistics Multiple RPׯ?R Square-knx`?Adjusted R SquareqV?Standard Error6Jq@ Observations~ 2@ ANOVA   df SS MS F Significance F   Regression~ ? mPM@ mPM@ ,@ Yms<  Residual~ 0@ JAUt@ JAU4@  Total 1@@  CoefficientsStandard Errort StatP-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Interceptt%“9d̃@0@F#;?`7@`7@ X (Machines)DxRy-@x,?w9$b<@yms<ecF+@O/@ecF+@O/@RESIDUAL OUTPUT ObservationPredicted Y (minutes) ResidualsStandard Residuals~ ?[ (@Pl}0TnЈ῾~ @[ (@IAU@u(?~ @K';@PYa@7TU‚?~ @K';@`M\"</߿~ @IAD@p.$8%I忾~ @mPL@@xRy@d _v?~ @mPL@[ /Ÿ꿾~  @mPL@o2y<| 5<A?HE?,p}[>6XXXXXXX T0! 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M43*43" 44%  SM03OU&$Q  Histogram'44eee >@    dMbP?_*+%MHP LaserJet 4000 PSW odXLetterPRIV''''b"dX??U} } m  X (Machines)T0 v v v v vvvv~ v&v Y (minutes) X (Machines) Residuals$@?Pl}0T1@?IAU@@@@PYa@9@@`M\"<C@@p.$O@@@xRy@J@@[ H@@o S@@ o@ R@@ [ @ @P@@ @xRy Q@@ 'ɟ׿ Q@@  U@@4=l}@X@@8@Y@@t%?@Z@@c.@]@ @@l}0T@ residual ~  ~  ~ @ ~ @   6G((((((((((((((((((#P ( \!wp!w!w!w  p  6NMM? @]`v  "??3` /Z` /Z?p3d23 M NM4 3Q:  ResidualsQ ;Q ;Q3_4E4D $% M 3O&Q4$% M 3O&Q4FAM8 ' 3OSP w 3*#M43*#M! M4523  O43d"  ,A3O ,% M3OQ443_ M NM  MM<444% L_M3O=Q'44eee >@ @Xp  Larry Winnerx Larry WinnerxMicrosoft Excel@Q|՜.+,D՜.+,L PXp x UF Statisticsma   Sheet2Sheet1Sheet3Sheet4Sheet6Sheet7Sheet5Sheet8 DocumentSummaryInformation8(_1125403002 FuǂXuǂOle CompObjbcalcmainChart1  Worksheets Charts 6> _PID_GUIDAN{64B84794-AF18-4247-B181-001DAAF513CB} FMicrosoft Excel ChartBiff8Excel.Sheet.89qObjInfoWorkbookœSummaryInformation(DocumentSummaryInformation8@      !"#$%&'()*+,-./01235[789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXY\^_`abcdefghijklmnopqrstuvwxyz{|}~  Ba= f =Kj8X1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1'_Arial1'_Arial1c_Arial1'_Arial1'_Arial1_Arial1_Arial1_Arial1_Arial11_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial1_Arial"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)                + ) , *   p  p "x@ &x@  t GChart1Sheet2(#Sheet1&Sheet31Sheet9 7Sheet10NSheet4`Sheet6jSheet7uSheet5Sheet8ccalcmain 2  P 3  @@  B( Y (minutes) X (Machines)BinMore FrequencySUMMARY OUTPUTRegression Statistics Multiple RR SquareAdjusted R SquareStandard Error ObservationsANOVA RegressionResidualTotal InterceptdfSSMSFSignificance F Coefficientst StatP-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%RESIDUAL OUTPUT ObservationPredicted Y (minutes) ResidualsStandard Residuals-7.5residual percentilez(pct) expected "3 x !N ' Vw%@J5??@ pdT0 =v@BbT0D3lw@B0Dx   Ir0=v=vM( MM=vS0t00533*^n00 00Ԣ00=vtb0:vPercent0]w&5,dw&54vp@Ν0?lbT03 owbT03wbT0<;wwHm+w4owwE w@,|0v<v~0BT0Lv`vu0(1T0|0v0v 0E 0#00#  "pip Squ?? 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DA(%0V@ Q@Dc.@~ .@IAU@0M4iҤ?D?@2@?DA(%06V@Q@DIAU@~ 0@@xRy@0[lٲe?D?@2@ ?DA(%` @Q@D@xRy@~ 1@PYa@0iҤI&?D?@2@?DA(%!@Q@DPYa@~ 2@o@0wܹs?D?@2@D%?DA(%mR @Q@Do@*rh ( `  p  6NMM? pi]`v  "??3` /Z'` /Z(?p3d23 M NM4 3Q:  ResidualsQ ;Q ;Q3_4E4D $% M 3O&'Q4$% M 3O&(Q4FA8 ' 3OSP w 3*#M43*#M! M4523  O43d"  ,A3O ,% M3OQ443_ M MM  MM<444% L_M`c3O=Q'44eee   >@    dMbP?_*+%MHP LaserJet 4000 PSW odXLetterPRIV''''b"dX??U} } m}  } $}  *  T0            v t  "     w    g   T  & wG G wT SUMMARY OUTPUTRegression Statistics Multiple RPׯ?R Square-knx`?Adjusted R SquareqV?Standard Error6Jq@ Observations~ 2@ ANOVA   df SS MS F Significance F   Regression~ ? mPM@ mPM@ ,@ Yms<  Residual~ 0@ JAUt@ JAU4@  Total 1@@  CoefficientsStandard Errort StatP-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Interceptt%“9d̃@0@F#;?`7@`7@ X (Machines)DxRy-@x,?w9$b<@yms<ecF+@O/@ecF+@O/@RESIDUAL OUTPUT ObservationPredicted Y (minutes) ResidualsStandard Residuals~ ?[ (@Pl}0TnЈ῾~ @[ (@IAU@u(?~ @K';@PYa@7TU‚?~ @K';@`M\"</߿~ @IAD@p.$8%I忾~ @mPL@@xRy@d _v?~ @mPL@[ /Ÿ꿾~  @mPL@o2y<| 5<A?HE?,p}[>6XXXXXXX T0! 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M43*43" 44%  JM03OU&Q  Histogram'44eee >@    dMbP?_*+%"??UColumn AT0 residual Frequency?@@@@@)@ More~ Dx+P(  p  6NM@? ]`%v  MHP LaserJet 4000 PSW odXLetterPRIV''''b"dX??3` o ` o ` o ` o  ` o !п 3d23 M NM4 3Q  FrequencyQ ;Q ;Q3_4E4D $% M 3O&Q4$% M 3O&Q4FA@g` 3OuW+ 3 b43*4% X K8M3O7&!Q residual'4% uvMZ3OC& Q  Frequency'4523  O43"  #`3O #% M3OQ4444A@g` 3O3 b! M43*43" 44%  JM03OU&Q  Histogram'44eee >@      dMbP?_*+%"??UColumn AT0T0 residual Frequency?@@@@@ More~ d+`(  p  6NM@?0 P]`v  "??3` o "` o #` o $` o %` o &@"?3d23 M NM4 3Q  FrequencyQ ;Q ;Q3_4E4D $% M 3O&"Q4$% M 3O&#Q4FA3Os-2 > 3 b43*4%  l H~M3O7&&Q residual'4% uXMZ3OC&%Q  Frequency'4523  O43"  []3O % M3OQ4444A3O3 b! 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