An antique collector believes that the price received for a particular item incr
ID: 3183036 • Letter: A
Question
An antique collector believes that the price received for a particular item increases with its age and with the number of bidders. He has collected a dataset of 32 recently auctioned comparable items. The variables in the dataset are Auction Price (in $), Age (in years), and Number of Bidders registered for the auction. Three regressions were run (see p. 8 for the output).
Which model provides the best fit to the data? How do you know this?
What is the proportion of variation in Auction Price explained by Number Bidders in Model A?
In Model B, we find that the P-value corresponding to the Constant is 0.4733. Explain what conclusion we can reach from this. Contrast with the P-value <0.0001 for Age of Item.
Provide a complete and precise interpretation of Number Bidders in Model C.
The collector would like a test of the claim that, in the population, every additional bidder increases the price by more than $50 (when controlling for Age). Conduct the appropriate test using Model C.
The collector believes that every additional year of age has a greater effect on prices of older items (whereas for newer items an additional year of age has a relatively small effect on price). Describe how this conjecture could be tested. [Does not require a numerical answer.]
To detect outliers an analyst decided to examine all those observations with unusually large and small values of Age of Item and Number Bidders. Would you agree with this approach? Why or why not?
Model A:
Multiple Regression for Auction Price
Summary
Multiple
R
R-Square
Adjusted
R-Square
StErr of
Estimate
0.3946
0.1557
0.1276
367.1969858
Degrees of
Freedom
Sum of
Squares
Mean of
Squares
F-Ratio
p-Value
ANOVA Table
Explained
1
746185.4264
746185.4264
5.5341
0.0254
Unexplained
30
4045008.792
134833.6264
Coefficient
Standard
Error
t-Value
p-Value
Regression Table
Constant
806.4049256
230.684572
3.4957
0.0015
Number Bidders
54.63620453
23.22502811
2.3525
0.0254
Model B:
Multiple Regression for Auction Price
Summary
Multiple
R
R-Square
Adjusted
R-Square
StErr of
Estimate
0.7302
0.5332
0.5177
273.0283995
Degrees of
Freedom
Sum of
Squares
Mean of
Squares
F-Ratio
p-Value
ANOVA Table
Explained
1
2554859.011
2554859.011
34.2729
< 0.0001
Unexplained
30
2236335.207
74544.50691
Coefficient
Standard
Error
t-Value
p-Value
Regression Table
Constant
-191.6575698
263.8865984
-0.7263
0.4733
Age of Item
10.47909492
1.789979792
5.8543
< 0.0001
Model C:
Multiple Regression for Auction Price
Summary
Multiple
R
R-Square
Adjusted
R-Square
StErr of
Estimate
0.9448
0.8927
0.8853
133.1365018
Degrees of
Freedom
Sum of
Squares
Mean of
Squares
F-Ratio
p-Value
ANOVA Table
Explained
2
4277159.703
2138579.852
120.6511
< 0.0001
Unexplained
29
514034.5153
17725.32812
Coefficient
Standard
Error
t-Value
p-Value
Regression Table
Constant
-1336.722052
173.3561261
-7.7108
< 0.0001
Age of Item
12.73619884
0.902380487
14.1140
< 0.0001
Number Bidders
85.8151326
8.705756815
9.8573
< 0.0001
Multiple Regression for Auction Price
Summary
Multiple
R
R-Square
Adjusted
R-Square
StErr of
Estimate
0.3946
0.1557
0.1276
367.1969858
Degrees of
Freedom
Sum of
Squares
Mean of
Squares
F-Ratio
p-Value
ANOVA Table
Explained
1
746185.4264
746185.4264
5.5341
0.0254
Unexplained
30
4045008.792
134833.6264
Coefficient
Standard
Error
t-Value
p-Value
Regression Table
Constant
806.4049256
230.684572
3.4957
0.0015
Number Bidders
54.63620453
23.22502811
2.3525
0.0254
Explanation / Answer
An antique collector believes that the price received for a particular item increases with its age and with the number of bidders. He has collected a dataset of 32 recently auctioned comparable items. The variables in the dataset are Auction Price (in $), Age (in years), and Number of Biddersregistered for the auction. Three regressions were run (see p. 8 for the output).
Here dependent variable is price
and independent variables are age and number of bidder registered for the auction.
Which model provides the best fit to the data? How do you know this?
Here for this question :
Calculate correlation coefficient between independent and dependent variables.
Take absolute value of correlation coefficient.
Calculate critical value using Pearson correlation coefficient table with df = n-2 and alpha value (0.05,0.01,0.1,….)
If | r | > Critical value model fits well or good.
And if | r | < Critical value model doesn’t fits good.
Here for three model we have to check the model fits well or not.
Model A :
R-sq = 0.1557
r = sqrt(0.1557) = 0.3946
Here r has positive sign because coefficient of number bidder has positive sign.
Now we have to find critical value.
Here n = 32
Assume alpha = 0.05
df = n-2 = 32-2 = 30
Critical value = 0.306
Here |r| > critical value
Model A fits well or good.
Model B :
R-sq = 0.5332
r = sqrt(0.5332) = 0.7302
Here r has positive sign because coefficient of age of item has positive sign.
Now we have to find critical value.
Here n = 32
Assume alpha = 0.05
df = n-2 = 32-2 = 30
Critical value = 0.306
Here |r| > critical value
Model fits well or good.
Model C :
R-sq = 0.8927
r = sqrt(0.8927) = 0.9448
Here r has positive sign because coefficient of both age of item and number bidder has positive sign.
Now we have to find critical value.
Here n = 32
Assume alpha = 0.05
df = n-2 = 32-2 = 30
Critical value = 0.306
Here |r| > critical value
Model C fits well or good.
All the three models fits well to the data.
What is the proportion of variation in Auction Price explained by Number Bidders in Model A?
It can be expressed by R-sq
R-sq=0.1557
It can be expressed by proportion of variation in price which is explained by variation in number bidders.
In Model B, we find that the P-value corresponding to the Constant is 0.4733. Explain what conclusion we can reach from this. Contrast with the P-value <0.0001 for Age of Item.
Here we can test the hyppothesis that,
H0 : B0 = 0 Vs H1 : B0 not=0
where B0 is the population intercept.
Assume alpha = level of significance = 0.05
Here test statistic follows t-distribution with n-2 degrees of freedom.
Here test statistic = -0.7263
And P-value = 0.4733
P-value > alpha
Accept H0 at 5% level of significance.
Conclusion : The population intercept may be 0.
Provide a complete and precise interpretation of Number Bidders in Model C.
In model C the regrssion equation is,
price = -1336.72 + 12.7362*age of item + 85.8151*number of bidders
Here if we fix age of item then one unit change in number of bidders will be 85.8151 increase in price.