EXST7034 – Regression Techniques Fall 2005 Geaghan Homework 1 Answer sheet Page 1 The SAS program I used to obtain the analyses for my answers is given below. A1 : Question 1.1 in KNNL) The example given was for sales dollar volume on the number of units sold. If there was no source or errors, this would be a functional relationship, such that Y = 2X If, however, there were clerical errors in sales, the relationship would no longer be perfectly fitted by this relationship. Although we may feel we know the underlying functional relationship, there would be uncertainty in the fit of the relationship. We would then fit the relationship using Y = β0 + β1 X + εi for i = 1, 2, , n where, Y = Dollar value of the sale X = Number of units sold The random error term εi would address the uncertainty, and give the variation due to clerical errors. Additionally, we might expect β0 to not differ significantly from 0, giving the relationship Y = β1X + εi . We may also hypothesize that β1 does not differ significantly from 2, if we feel that there is no bias or consistent tendency in the so called “clerical errors”, then the relationship is Y = 2X + εi . A2 : Question 1.2 in KNNL) This function would be would be fixed at Y = 300 + 2X, and would be a functional relationship barring any “clerical errors”. B1 : Question 1.20a in KNNL) Obtain estimated regression function Yi =–0.58016 + 15.03525Xi. Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 -0.58016 2.80394 -0.21 0.8371 machines 1 15.03525 0.48309 31.12 <.0001dm'log;clear;output;clear'; *************************************************************; *** EXST7034 Homework Example 1 ***; *** Problem from Neter, Wasserman & Kuttner 1989, #2.18 ***; *************************************************************; OPTIONS LS=132 PS=256 NOCENTER NODATE NONUMBER nolabel; filename copier 'C:\Geaghan\Current\EXST7034\Fall2005\SAS\CH01PR20.txt'; ODS HTML style=minimal rs=none body='C:\Geaghan\Current\EXST7034\Fall2005\SAS\CH01PR20a.html' ; Title1 'Assignment 1 : Copier maintainance example'; DATA ONE; INFILE copier MISSOVER; LABEL machines = 'Number of machines serviced'; LABEL minutes = 'Minutes to service machines'; INPUT minutes machines; CARDS; RUN; ; options ps=45; PROC PLOT DATA=ONE; PLOT minutes * machines; run; options ps=256; PROC REG DATA=ONE lineprinter; ID machines; MODEL minutes = machines / XPX I P; run; options ps=55; plot predicted.*machines='X' minutes*machines='o' / overlay; output out=next1 p=yhat r=e; run; options ps=256; proc sort data=next1; by machines; run; proc print data=next1; run; PROC REG DATA=ONE; MODEL minutes = machines; restrict intercept = 0; run; PROC GLM; MODEL minutes = machines / XPX I P; run;EXST7034 – Regression Techniques Fall 2005 Geaghan Homework 1 Answer sheet Page 2 B1 : Question 1.20b in KNNL) plot reg function and data together Looks pretty good to me. Notice “X” has been used for the predicted values and “o” for the observed. The questions marks denote where SAS had to place BOTH a “X” and an “o”. B1 : Question 1.20c in KNNL) give an interpretation for b0 The regression function is Yi = –0.58016 + 15.03525Xi. The intercept is theoretical amount of time required to service a machine when no machine is serviced. We do not know how this company bills for “service time”. This value could include travel time, time needed for setting up equipment before actually working on a machine, time to do paperwork after working on a machine. If any of these are included in the time for a call, then we would expect the intercept to be greater than zero, and it would estimated the time needed (in minutes) for these addition tasks. If on the other hand the “service time” includes only time spent working on a machine then we would expect the intercept to be zero (e.g. no machine serviced requires no time). What we actually observe is a negative number. If the real value is zero then we expect to see a small positive number about half the time and a small negative number about half the time. The question then becomes, does this number differ significantly from zero? SAS tests this (see table for question 1.20a above) and shows that the observed value does not differ significantly from zero (P>F= 0.8371). If the hypotheses was rejected (Ho: β0 = 0) and the value was negative then we may want to question the model adequacy. The bottom line: YES, I would say that the intercept does tell us something about how this company works and counts the time recorded as “service time”. It may also tell us something about the best model (which probably should go through the origin). --------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-------- PRED | | 160 + + | o | | | | X | | o | 140 + + P | ? o | r | o o | e | o | d |