Pricing Diamond Rings

Pricing diamond rings in Singapore can be viewed as an interesting exercise in 
statistical modelling. The price equals the current market value of the gold 
content of the ring, a craftsmanship fee plus the cost of the diamond. The 
price of a diamond depends upon the four Cs: caratage, cut, colour and clarity. 
For jewelry intended for the mass market it can be assumed that the major 
factor in determining price is the caratage, i.e. the size of the diamond.

On February 29, 1992 a full page ad was placed in the Singapore Straits Times 
newspaper. The advertisement contained pictures of diamond rings and listed 
their prices, the caratage of the diamond and the gold purity. The data below 
is for 20 carat gold ladies' rings, each mounted with a single diamond.

As only the caratage of the diamond and the purity of the gold was stated in the 
ad, it can be assumed that the amount of gold in each ring and the craftsmanship 
fee varied little between rings, and the cut, colour and clarity of each diamond 
was also similar. Hence we will consider only the size of the diamond in the 
ring and its cost and attempt to find the mathematical function that best fits 
these data.

There were 48 rings of varying designs, with the diamonds weighing from 0.12 to 
0.35 carats (one carat = 0.2 gram) and priced between $223 and $1086. The data 
are given below.

Carats 	Cost($) 	Carats 	Cost($)
0.17 	355 	0.17 	353
0.16 	328 	0.18 	438
0.17 	350 	0.17 	318
0.18 	325 	0.18 	419
0.25 	642 	0.17 	346
0.16 	342 	0.15 	315
0.15 	322 	0.17 	350
0.19 	485 	0.32 	918
0.21 	483 	0.32 	919
0.15 	323 	0.15 	298
0.18 	462 	0.16 	339
0.28 	823 	0.16 	338
0.16 	336 	0.23 	595
0.2 	498 	0.23 	553
0.23 	595 	0.17 	345
0.29 	860 	0.33 	945
0.12 	223 	0.25 	655
0.26 	663 	0.35 	1086
0.25 	750 	0.18 	443
0.27 	720 	0.25 	678
0.18 	468 	0.25 	675
0.16 	345 	0.15 	287
0.17 	352 	0.26 	693
0.16 	332 	0.15 	316

 
1. Find a linear model for this data. Discuss. What is the y-intercept of
   your model? Is this sensible? Does it help if we force the data through
   the origin?
2. Larger diamonds may not fit a linear pattern as they are quite rare. Given
   this and the non-zero y-intercept of the linear model, it is worth  
   determining if a non-linear model better fits the data.

Test each of the following nonlinear models:

y = a*x2

y= a*eb*x

y = a*eb*x^2 + c*x + d

In each case, determine how well the model fits the existing data, and if it 
extrapolates sensibly for larger and smaller sizes of diamonds.