Let's solve the problem step by step.
First, we need to understand the concept of residuals. The residuals are the differences between the actual sales and the predicted sales. For each month, the residual can be calculated as:
[tex]\[ \text{Residual} = \text{Actual Sales} - \text{Predicted Sales} \][/tex]
We are given the following data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
& \text{Month 1} & \text{Month 2} & \text{Month 3} & \text{Month 4} & \text{Month 5} & \text{Month 6} \\
\hline
\text{Actual Sales} & 55 & 150 & 325 & 510 & 780 & 990 \\
\hline
\text{Predicted Sales} & 40 & 150 & 300 & 500 & 800 & 1,000 \\
\hline
\end{array}
\][/tex]
We will calculate the residuals for each month:
- For Month 1:
[tex]\[
\text{Residual} = 55 - 40 = 15
\][/tex]
- For Month 2:
[tex]\[
\text{Residual} = 150 - 150 = 0
\][/tex]
- For Month 3:
[tex]\[
\text{Residual} = 325 - 300 = 25
\][/tex]
- For Month 4:
[tex]\[
\text{Residual} = 510 - 500 = 10
\][/tex]
- For Month 5:
[tex]\[
\text{Residual} = 780 - 800 = -20
\][/tex]
- For Month 6:
[tex]\[
\text{Residual} = 990 - 1,000 = -10
\][/tex]
We found the residuals to be:
[tex]\[
[15, 0, 25, 10, -20, -10]
\][/tex]
Next, we sum all these residuals to find the sum of the residuals.
[tex]\[
\text{Sum of Residuals} = 15 + 0 + 25 + 10 + (-20) + (-10)
\][/tex]
Adding these values step by step:
[tex]\[
15 + 0 = 15
\][/tex]
[tex]\[
15 + 25 = 40
\][/tex]
[tex]\[
40 + 10 = 50
\][/tex]
[tex]\[
50 - 20 = 30
\][/tex]
[tex]\[
30 - 10 = 20
\][/tex]
Thus, the sum of the residuals is:
[tex]\[
20
\][/tex]
The correct answer is:
[tex]\[
\boxed{20}
\][/tex]
