Philco Diamonds buys diamonds from the Canadian Arctic and sells them in Dubai. Management has been confused by the prices they must pay for the diamonds they buy.Sometimes the price seems too high, sometimes too low. They cannot detect a pattern to help them determine a fair buying price.

You investigate the problem and determine that diamonds are commonly valued according to "4 C's"....that is carats, color, cut and clarity defined as follows:

Carat: the number of carat weight of the diamond (more is better)

Color: the color of the diamond on a scale of 1 to 10 (10 being best)

Cut: the cut of the diamond on a scale of 1 to 5 (5 being best)

Clarity: the clarity of the diamond on a scale of 1 to 5 (5 being best)

You investigate Philco records and find the following data about the last 20 diamonds bought: (this data is available in Excel format on Moodle)

Diamond #	Carat	Color	Cut	Clarity	Price
1	4.3	1	3	5	37,715
2	1.3	4	4	1	20,795
3	1.8	6	1	2	14,680
4	1.5	4	3	4	55,348
5	3.6	2	1	2	7,121
6	1.9	9	5	5	163,728
7	2.4	3	1	5	88,877
8	4.6	2	3	1	5,211
9	4.6	6	3	2	42,227
10	1.7	7	2	3	33,805
11	2.4	6	2	1	28,543
12	3.6	1	1	4	3,993
13	1.5	6	4	1	4,516
14	4.7	4	2	2	66,183
15	2.0	5	4	5	145,500
16	4.7	4	3	1	10,297
17	4.8	5	4	2	86,719
18	2.6	3	1	3	21,755
19	2.4	1	3	1	3,038
20	2.3	10	5	3	58,424

a) State the multiple regression equation in specific terms.

b) Is the model the best we can do given the data we have available?

c) Does the goodness of fit measurement indicate this model is good enough to be used? Explain.

d) Manually calculate the t-value for color. Show your work.

e) Does the intercept amount seem reasonable to you?Explain referring to the relevant ranges in this example.

f) Using the best model you have, estimate the selling price of a diamond having the following characteristics:

Carat = 1.2 Color = 5 Cut = 3 Clarity = 3

g) Using the best model you have, estimate the selling price of a diamond having the following characteristics:

Carat = 2.9 Color = 9 Cut = 4 Clarity = 3

h) Comment on the t-values of the 4 independent variables. Explain what this means.

i) Run a test to determine what the approximate combined effect of the 4 X-variables is, that is what is an estimate of a combined t-value? Comment on your results.

j) Comment of the Economic Plausibility of this example.

k) Comment on the homoscedasticity of each of the 4 independent variable residuals. Explain what this means.

l) Comment on the independence of each of the 4 independent variable residuals. Explain what this means.

m) Comment on the normality of the residuals. Explain what this means.

Philco Diamonds:
Diamond #	Carat	Color	Cut	Clarity	Price
1	4.3	1	3	5	37,715
2	1.3	4	4	1	20,795
3	1.8	6	1	2	14,680
4	1.5	4	3	4	55,348
5	3.6	2	1	2	7,121
6	1.9	9	5	5	163,728
7	2.4	3	1	5	88,877
8	4.6	2	3	1	5,211
9	4.6	6	3	2	42,227
10	1.7	7	2	3	33,805
11	2.4	6	2	1	28,543
12	3.6	1	1	4	3,993
13	1.5	6	4	1	4,516
14	4.7	4	2	2	66,183
15	2	5	4	5	145,500
16	4.7	4	3	1	10,297
17	4.8	5	4	2	86,719
18	2.6	3	1	3	21,755
19	2.4	1	3	1	3,038
20	2.3	10	5	3	58,424