To determine whether the equation that produced the predicted values represents a good line of best fit, we need to analyze the residuals. The residuals are the differences between the actual values and the predicted values. In this case:
- Actual values: [tex]\([12, 14, 14, 13, 16, 14]\)[/tex]
- Predicted values: [tex]\([8, 15, 15, 12, 17, 15]\)[/tex]
- Residuals: [tex]\([4, -1, -1, 1, -1, -1]\)[/tex]
A good line of best fit will have residuals that are close to zero and a sum of residuals that is also close to zero.
Let’s evaluate this step-by-step:
1. Calculate the residuals:
- January: [tex]\(12 - 8 = 4\)[/tex]
- February: [tex]\(14 - 15 = -1\)[/tex]
- March: [tex]\(14 - 15 = -1\)[/tex]
- April: [tex]\(13 - 12 = 1\)[/tex]
- May: [tex]\(16 - 17 = -1\)[/tex]
- June: [tex]\(14 - 15 = -1\)[/tex]
2. Check the individual residuals:
[tex]\(4, -1, -1, 1, -1, -1\)[/tex]
3. Examine how close each residual is to zero:
- In this case, not all residuals are close to zero. For instance, [tex]\(4\)[/tex] is relatively far from zero.
4. Sum the residuals:
[tex]\[
4 + (-1) + (-1) + 1 + (-1) + (-1) = 1
\][/tex]
5. Analyze the sum of the residuals:
- While the sum of the residuals is [tex]\(1\)[/tex], which is a small number, the question also asks whether the residuals are all close to zero, which they are not.
Therefore, the conclusion is:
The equation is not a good fit because the residuals [tex]\( [4, -1, -1, 1, -1, -1] \)[/tex] are not all close to zero. Therefore, the best choice from the given options is:
- No, the equation is not a good fit because the residuals are all far from zero.
