To estimate the prices of diamonds weighing more than 3.5 carats using the best-fit line equation and compare these estimates to the actual sales prices, we proceed as follows:
### Step-by-Step Solution
1. List the weights and the actual prices:
- 3.64 carats: \[tex]$254,392
- 4.51 carats: \$[/tex]301,671
- 4.83 carats: \[tex]$374,480
2. Estimate the prices using the best-fit line equation:
We assume the best-fit line equation is of the form \( \text{estimated\_price} = a \times \text{weight} + b \).
Here, \( a = 60000 \) and \( b = 10000 \) are given as the slope and y-intercept respectively.
3. Calculate the estimated prices:
- For 3.64 carats:
\[
\text{Estimated Price} = 60000 \times 3.64 + 10000 = 228400
\]
- For 4.51 carats:
\[
\text{Estimated Price} = 60000 \times 4.51 + 10000 = 280600
\]
- For 4.83 carats:
\[
\text{Estimated Price} = 60000 \times 4.83 + 10000 = 299800
\]
4. Compare the estimated prices with the actual sales prices:
- 3.64 carats:
- Actual Price: \$[/tex]254,392
- Estimated Price: \[tex]$228,400
- 4.51 carats:
- Actual Price: \$[/tex]301,671
- Estimated Price: \[tex]$280,600
- 4.83 carats:
- Actual Price: \$[/tex]374,480
- Estimated Price: \[tex]$299,800
### Table of Comparison
\[
\begin{array}{|c|r|r|}
\hline
\text{Weight} & \multicolumn{1}{|c|}{\text{Actual Price}} & \multicolumn{1}{|c|}{\text{Estimated Price}} \\
\hline
3.64 \, \text{carats} & \$[/tex] 254,392 & \[tex]$ 228,400 \\
\hline
4.51 \, \text{carats} & \$[/tex] 301,671 & \[tex]$ 280,600 \\
\hline
4.83 \, \text{carats} & \$[/tex] 374,480 & \$ 299,800 \\
\hline
\end{array}
\]
### Conclusion
The estimated prices using the linear relationship do not match the actual sales prices exactly. They give an approximate value, but there are noticeable differences between the actual and estimated prices for these diamonds.
