To determine the estimated prices of the diamonds weighing 4.51 carats and 4.83 carats and to compare these estimates with the actual sales prices, we use the equation of the best-fit line. This equation is in the form [tex]\( y = mx + b \)[/tex], where [tex]\( y \)[/tex] is the estimated price, [tex]\( x \)[/tex] is the weight of the diamond, [tex]\( m \)[/tex] is the slope, and [tex]\( b \)[/tex] is the y-intercept.
From the information provided:
[tex]\[ m = 53285.4478 \][/tex]
[tex]\[ b = 0 \][/tex]
[tex]\[ \text{Estimated price} = m \times \text{weight} + b \][/tex]
### Step-by-Step Solution:
1. Estimate the price for 4.51 carats:
[tex]\[ \text{Estimated price} = 53285.4478 \times 4.51 \][/tex]
[tex]\[ \text{Estimated price} \approx 240317.37 \][/tex]
2. Estimate the price for 4.83 carats:
[tex]\[ \text{Estimated price} = 53285.4478 \times 4.83 \][/tex]
[tex]\[ \text{Estimated price} \approx 257368.71 \][/tex]
### Completed Table:
\begin{tabular}{|l|r|l|}
\hline \multicolumn{2}{|c|}{} \\
\hline Weight & Actual Price & \begin{tabular}{c}
Estimated Price: \\
Linear Relationship
\end{tabular} \\
\hline 3.64 carats & \[tex]$254,392 & 193,959.03 \\
\hline 4.51 carats & \$[/tex]301,671 & 240,317.37 \\
\hline 4.83 carats & \[tex]$374,480 & 257,368.71 \\
\hline
\end{tabular}
### Comparison of Estimates with Actual Prices:
- For 3.64 carats:
\[ \text{Actual Price}: \$[/tex]254,392 \]
[tex]\[ \text{Estimated Price}: \$193,959.03 \][/tex]
The difference between the actual price and estimated price is \[tex]$60,432.97. Clearly, the estimated price is significantly lower than the actual price.
- For 4.51 carats:
\[ \text{Actual Price}: \$[/tex]301,671 \]
[tex]\[ \text{Estimated Price}: \$240,317.37 \][/tex]
The difference between the actual price and estimated price is \[tex]$61,353.63. Here as well, the estimated price is lower than the actual price.
- For 4.83 carats:
\[ \text{Actual Price}: \$[/tex]374,480 \]
[tex]\[ \text{Estimated Price}: \$257,368.71 \][/tex]
The difference between the actual price and estimated price is \$117,111.29. Again, the estimated price is less than the actual price by a considerable margin.
### Conclusion:
The estimates do not closely match the actual sales prices, with the estimated prices generally being significantly lower for each corresponding weight. This discrepancy suggests that the simple linear model [tex]\( y = 53285.4478x \)[/tex] may not fully capture the relationship between weight and price for these diamonds. Factors other than just weight likely play a significant role in the actual pricing of diamonds.
