Answered

Get comprehensive solutions to your questions with the help of IDNLearn.com's experts. Find reliable solutions to your questions quickly and accurately with help from our dedicated community of experts.

An equation was created for the line of best fit from the actual enrollment data. It was used to predict the dance studio enrollment values shown in the table below:

\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Enrollment} & \multicolumn{6}{|c|}{Month} \\
\cline{2-7}
& January & February & March & April & May & June \\
\hline
Actual & 120 & 140 & 150 & 140 & 150 & 130 \\
\hline
Predicted & 80 & 150 & 110 & 150 & 110 & 150 \\
\hline
Residual & 40 & -10 & 40 & -10 & 40 & -20 \\
\hline
\end{tabular}

Analyze the data. Determine whether the equation that produced the predicted values represents a good line of best fit.

A. No, the equation is not a good fit because the residuals are all far from zero.
B. No, the equation is not a good fit because the sum of the residuals is a large number.
C. Yes, the equation is a good fit because the residuals are not all far from zero.
D. Yes, the equation is a good fit because the sum of the residuals is a small number.

Sagot :

GinnyAnsweridn
To determine whether the equation for the line of best fit provides a good prediction of the dance studio enrollment, let's analyze the given data step by step.

### Step 1: Residuals
The residuals are calculated as the difference between the actual values and the predicted values for each month:
- January: Actual = 120, Predicted = 80, Residual = [tex]\(120 - 80 = 40\)[/tex]
- February: Actual = 140, Predicted = 150, Residual = [tex]\(140 - 150 = -10\)[/tex]
- March: Actual = 150, Predicted = 110, Residual = [tex]\(150 - 110 = 40\)[/tex]
- April: Actual = 140, Predicted = 150, Residual = [tex]\(140 - 150 = -10\)[/tex]
- May: Actual = 150, Predicted = 110, Residual = [tex]\(150 - 110 = 40\)[/tex]
- June: Actual = 130, Predicted = 150, Residual = [tex]\(130 - 150 = -20\)[/tex]

Summarizing the residuals:
[tex]\[ [40, -10, 40, -10, 40, -20] \][/tex]

### Step 2: Sum of Residuals
The sum of the residuals is:
[tex]\[ 40 + (-10) + 40 + (-10) + 40 + (-20) = 80 \][/tex]

### Step 3: Mean of Residuals
To find the mean of the residuals:
[tex]\[ \text{Mean of Residuals} = \frac{\text{Sum of Residuals}}{\text{Number of Residuals}} = \frac{80}{6} \approx 13.33 \][/tex]

### Step 4: Analyzing Residuals
1. Residuals Far from Zero:
We check if all the residuals are far from zero. Typically, a residual is considered "far from zero" if its absolute value is greater than 5 or another threshold defined by the context.
[tex]\[ |40| > 5, |-10| > 5, |40| > 5, |-10| > 5, |40| > 5, |-20| > 5 \][/tex]
All these residuals have an absolute value greater than 5, hence they are all considered far from zero.

2. Large Sum of Residuals:
A large sum of residuals indicates that the total error across all predictions is significant. Generally, we might define "large" based on the context or a fixed threshold, but given the sum here (80), it is quite clear that it is not small.

### Conclusion
Based on our analysis:
- The residuals are all far from zero.
- The sum of the residuals is a large number.

Therefore, both criteria indicate that the equation that produced the predicted values does not represent a good line of best fit. The correct conclusion is:
- No, the equation is not a good fit because the residuals are all far from zero.
- No, the equation is not a good fit because the sum of the residuals is a large number.
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.