An art collector is studying the relationship between the selling price of a pai

Question

An art collector is studying the relationship between the selling price of a painting and two independent variables. The two independent variables are the number of bidders at the particular auction and the age of the painting, in years. A sample of 25 paintings revealed the following sample information.

a-1. Develop a multiple regression equation using the independent variables number of bidders and age of painting to estimate the dependent variable auction price. (Round your answers to 2 decimal places.)

a-2. Complete the following table: (Negative amounts should be indicated by a minus sign. Round your answers to 2 decimal places.)

a-3. Discuss the equation. Does it surprise you that there is an inverse relationship between the number of bidders and the price of the painting? (Round your answers to 1 decimal place.)

b-1. Create an interaction variable and include it in the regression equation. (Negative answers should be indicated by a minus sign. Round your answers to 2 decimal places.)

b-2. Complete the following table: (Negative answers should be indicated by a minus sign. Round your answers to 3 decimal places.)

b-3. What is the corresponding t value to the interaction term? (Round your answer to 2 decimal places.)

Painting Auction Price Bidders Age Painting Auction Price Bidders Age 1 3,470 10 67 14 4,020 6 79 2 3,500 8 56 15 4,190 4 83 3 3,700 7 73 16 4,130 3 71 4 3,860 4 71 17 4,130 9 89 5 3,920 12 99 18 4,370 5 103 6 3,900 10 87 19 4,450 3 106 7 3,830 11 78 20 4,390 8 93 8 3,940 8 83 21 4,380 8 88 9 3,880 13 90 22 4,540 4 96 10 3,940 13 98 23 4,660 5 94 11 4,200 0 91 24 4,710 3 88 12 4,060 7 93 25 4,880 1 84 13 4,200 2 97

Explanation / Answer

## By using R-software

> Selling=scan("clipboard")

Read 25 items
> Bidders=scan("clipboard")
Read 25 items
> Age=scan("clipboard")
Read 25 items
> fit=lm(Selling~Bidders+Age)
> fit

Call:
lm(formula = Selling ~ Bidders + Age)

Coefficients:
(Intercept) Bidders Age  
3080.05 -54.19 16.29  

> summary(fit)

Call:
lm(formula = Selling ~ Bidders + Age)

Residuals:
Min 1Q Median 3Q Max
-362.36 -155.62 -31.93 87.91 485.85

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 3080.053 343.885 8.957 8.62e-09 ***
Bidders -54.189 12.281 -4.412 0.000220 ***
Age 16.289 3.784 4.305 0.000287 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 222.1 on 22 degrees of freedom
Multiple R-squared: 0.6478, Adjusted R-squared: 0.6158
F-statistic: 20.23 on 2 and 22 DF, p-value: 1.034e-05

a1) The multiple regression model is

Selling = 3080.05 -54.19*Bidders + 16.29*Age

a3) From the above multiple regression model there is an inverse relationship between the Selling price (Auction price) and the number of bidders. A unit increse in the number of bidders there is change in auction price by 54.19 dollars.

b1) The regression model when the interaction term of bidders and age is included in the model is:

Selling= 3971.678-184.994*Bidders+6.353*Age+1.462*Bidders*Age

b3) The t value for interaction term is 1.145 with pvalue 0.265015 which indicates that the interaction is not significant in the model at 5% level of significance..

> fit=lm(Selling~Bidders+Age+Bidders*Age)
> fit

Call:
lm(formula = Selling ~ Bidders + Age + Bidders * Age)

Coefficients:
(Intercept) Bidders Age Bidders:Age  
3971.678 -184.994 6.353 1.462  

> summary(fit)

Call:
lm(formula = Selling ~ Bidders + Age + Bidders * Age)

Residuals:
Min 1Q Median 3Q Max
-349.80 -112.06 -49.19 136.96 436.85

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) 3971.678 850.175 4.672 0.000131 ***
Bidders -184.994 114.871 -1.610 0.122230   
Age 6.353 9.455 0.672 0.508962   
Bidders:Age 1.462 1.277 1.145 0.265015   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 220.5 on 21 degrees of freedom
Multiple R-squared: 0.6685, Adjusted R-squared: 0.6211
F-statistic: 14.12 on 3 and 21 DF, p-value: 2.916e-05

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