Question

1. Partial F Test. For this and the next 2 parts: A collector of antique grandfather clocks believes that the price received for the clocks at an antique auction increases with the age of the clock (X1) and with the number of bidders (X2). Suppose the collector, having observed many auctions, believes that the rate of increase of the auction price with age will be driven upward by a large number of bidders. As a result, the interaction term, X1X2, is added to the model. Also, two second-order terms, X1-SQ and X2-SQ, are included. Thus, the first order model is the "reduced model" while the second-order model is considered the "complete model." The models are summarized as follows:
   First-order model: E(Y) = B0 + B1X1 + B2X2
   Second-order model: E(Y) = B0 + B1X1 + B2X2 + B3X1X2 + B4X1-SQ + B5X2-SQ
   Since the second-order model contains all the terms of the first-order model in addition to additional terms, it is said that the two models are nested. You wish to test whether the second-order model contributes more information for the prediction of auction price. The regression output of the COMPLETE MODEL is given below.

   SUMMARY OUTPUT
   Regression Statistics
   Multiple R	0.9799
   R2	0.9602
   Adj. R2	0.9526
   SY.X	85.6221
   n	32
   ANOVA
   	df	SS	MS	F	P-value
   Regression	5	4600584	920116.9	125.5078	0.0000
   Residual	26	190609.9	7331.152
   Total	31	4791194
   	Coeff.	Std. Err.	t Stat	P-value	Lower 95%	Upper 95%
   Intercept	-340.0331	760.6677	-0.4470	0.6586	-1903.6091	1223.5428
   X1	3.4144	8.8971	0.3838	0.7043	-14.8739	21.7027
   X2	13.6289	61.8639	0.2203	0.8274	-113.5343	140.7921
   X1X2	1.1234	0.2303	4.8781	0.0000	0.6500	1.5968
   X12	-0.0037	0.0273	-0.1358	0.8930	-0.0599	0.0525
   X22	-4.1290	2.1331	-1.9357	0.0638	-8.5136	0.2555

   Part A.
   If the second-order terms contribute to the model, then

   		The sum of squares error for the reduced model (SSER) should be much smaller than the sum of squares error for the complete model (SSEC) in a significant way.
   		The sum of squares error for the complete model (SSEC) should be much smaller than the sum of squares error for the reduced model (SSER) in a significant way.
   		the difference, (SSER - SSEC), will be very small and insignificant
   		None of the above

Part B.

1. The sum of squares error for the REDUCED MODEL 514,034.5. Compute the F statistic to test the null hypothesis that the coefficients of the second-order terms are equal to zero in a partial F test.

   		14.71
   		0.91
   		7,331.15
   		2.98

Part C.

1. Would you conclude, at the 0.05 level, that the inclusion of the second-order terms contributes significantly to the prediction of antique clock prices?

   		Yes, F value from Exhibit 2 is significant
   		Yes, F value for the difference, (SSER - SSEC), is significant
   		Yes, F value from Exhibit 1 is significant
   		No, p value for the difference, (SSER - SSEC), is greater than 0.05

Answer 1

PART A) The sum of squares error for the reduced mode should be much smaller than the sum of square error for the complete model.

PART B) F test is 14.71

PART C) YES Fvalue for the difference is significant