Typically, the sales volume declines with an increase of a product price. It has

Question

Typically, the sales volume declines with an increase of a product price. It has been observed, however, that for some luxury goods the sales volume may increase when the price increases. The following Excel output illustrates this rather unusual relationship. Scatterplot of Price Vs Sales for a Luxury Good 2500 y 0.3224x + 1637.6 R1 =0.0012 y- 0.2702x-57.463+ 42123 #07991 y- 0.0007x+0.0352 34 581+3539.2 R0.8055 1500 Scatterplot of Price vs Sales for a Luxury Good Linear (Scatterplot of Price Vs Sales for a Luxury Good) Poly. (Scatterplot of Price Vs Sales for 300 Luxury Good) Poly. (Scatterplot of Price vs Sales for Luxury Good) 50 100 150 200 What is the number of estimated coefficients of the cubic regression model? 4

Explanation / Answer

(first part)right choice is (fourth) 4

the cubic equation is fitted as y=b0+b1x+b2x2+b3x3 ,

here there are four constant b0=3539.2 ,b1=-34.581,b2=0.0352,b3=0.0007 are estimated

so number of estimated coefficeints are 4, right choice is fourth 4

(second part) right choice is (second) quadratic

the explained variation (R2) by linear is 0.0012 ,by quadratic is 0.7991 and by cubic is 0.8055,

since difference of R2 is very high between linear and quadratic and difference of R2 is also very low between quadratic and cubic. it means adding quadratic term gives significant amout of increase in R2 but adding cubic term is not giving much increase in R2, so quadratic is better explaining

(third part) right choice is (first) 106.33

y=0.2702x2-57.463x+4212.2

dy/dx=0.2702*2*x-57.463

for maxima-minima put dy/dx=0

or, 0.2702*2*x-57.463=0

or, x=57.463/(0.2702*2)=106.33

required answer is 106.33

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