Diamonds are often thought of as a cherished item with a personal value well in excess of their monetary value.The monetary value of a diamond is determined by its exact quality as defined by the 4 C’s: cut, color, clarity, and carat weight. The price (dollars) and the carat weight of a diamond are its two most known characteristics. In order to understand the role carat weight has in determining the price of a diamond, the carat weight and price of 20 loose round diamonds, all of color D and clarity VS1, were obtained January 7, 2010, from the Internet.
Carat Wt Price
0.56....... 2055
0.90..... 5433
0.50..... 1735
0.53..... 1962
0.92..... 5554
0.51..... 1900
0.41..... 1264
0.40..... 1242
0.80..... 4182
0.57..... 2085
0.71..... 3117
0.40..... 1176
0.30..... 855
0.40..... 1153
0.62..... 2384
0.54..... 1746
0.30..... 894
0.50..... 1871
0.54..... 1746
0.70........ 3074
a. Draw a scatter diagram of the data: carat weight (x) and price (y).
b. Do the data suggest a linear relationship for the domain 0.30 to 0.92 carat? Discuss your findings in part a.
c. Diamonds smaller than 0.30 carat and diamonds larger than 0.92 carat may not fit the linear pattern demonstrated by this data. Explain.
d. Find the equation for the line of best fit.
e. According to this information, what would be a typical price for a 0.75-carat loose diamond of this quality?
f. On the average, by how much does the price increase for each extra 0.01 carat in weight? Within what interval of x values would you expect this to be true?
g. Find the variance of y about the regression line. What characteristics in the scatter diagram support this large value?