Answer :
We are given eight data points with coordinates corresponding to a month number (represented by [tex]$x$[/tex]) and the average monthly temperature (represented by [tex]$y$[/tex]):
[tex]$$
\begin{array}{cc}
x & y \\
\hline
1 & 58 \\
3 & 60 \\
4 & 64 \\
6 & 68 \\
7 & 70 \\
8 & 73 \\
9 & 71 \\
12 & 55 \\
\end{array}
$$[/tex]
We wish to determine which one of the four candidate regression equations best fits this data. The candidates are:
1. [tex]$$ y = 7.57 \sin(0.55x + 0.68) + 64.40 $$[/tex]
2. [tex]$$ y = 7.57 \sin(0.68x + 0.55) + 64.40 $$[/tex]
3. [tex]$$ y = 7.57 \sin(-0.55x - 0.68) + 64.40 $$[/tex]
4. [tex]$$ y = 7.57 \sin(-0.68x - 0.55) + 64.40 $$[/tex]
The approach is to compute the predicted temperature for each data point using each candidate equation and then compare the accuracy using the sum of squared errors (SSE). For each candidate, the error at a data point is given by
[tex]$$ \text{Error} = \left( y_{\text{predicted}} - y_{\text{actual}} \right)^2, $$[/tex]
and the SSE for a candidate is the sum of these squared errors over all data points.
After performing the calculations for each candidate, the computed overall errors (SSE) were found to be approximately:
- Candidate 1: [tex]$$\text{SSE}_1 \approx 1161.01$$[/tex]
- Candidate 2: [tex]$$\text{SSE}_2 \approx 854.98$$[/tex]
- Candidate 3: [tex]$$\text{SSE}_3 \approx 27.29$$[/tex]
- Candidate 4: [tex]$$\text{SSE}_4 \approx 168.82$$[/tex]
Since the goal is to minimize the squared error, the best candidate is the one with the smallest SSE. Here, Candidate 3 has the lowest SSE of about [tex]$27.29$[/tex], which indicates that it provides the best fit to the data.
Thus, the regression equation that best models the data is
[tex]$$
\boxed{y = 7.57 \sin(-0.55x - 0.68) + 64.40.}
$$[/tex]
This is the regression equation that most accurately represents the given table of average monthly temperatures.
[tex]$$
\begin{array}{cc}
x & y \\
\hline
1 & 58 \\
3 & 60 \\
4 & 64 \\
6 & 68 \\
7 & 70 \\
8 & 73 \\
9 & 71 \\
12 & 55 \\
\end{array}
$$[/tex]
We wish to determine which one of the four candidate regression equations best fits this data. The candidates are:
1. [tex]$$ y = 7.57 \sin(0.55x + 0.68) + 64.40 $$[/tex]
2. [tex]$$ y = 7.57 \sin(0.68x + 0.55) + 64.40 $$[/tex]
3. [tex]$$ y = 7.57 \sin(-0.55x - 0.68) + 64.40 $$[/tex]
4. [tex]$$ y = 7.57 \sin(-0.68x - 0.55) + 64.40 $$[/tex]
The approach is to compute the predicted temperature for each data point using each candidate equation and then compare the accuracy using the sum of squared errors (SSE). For each candidate, the error at a data point is given by
[tex]$$ \text{Error} = \left( y_{\text{predicted}} - y_{\text{actual}} \right)^2, $$[/tex]
and the SSE for a candidate is the sum of these squared errors over all data points.
After performing the calculations for each candidate, the computed overall errors (SSE) were found to be approximately:
- Candidate 1: [tex]$$\text{SSE}_1 \approx 1161.01$$[/tex]
- Candidate 2: [tex]$$\text{SSE}_2 \approx 854.98$$[/tex]
- Candidate 3: [tex]$$\text{SSE}_3 \approx 27.29$$[/tex]
- Candidate 4: [tex]$$\text{SSE}_4 \approx 168.82$$[/tex]
Since the goal is to minimize the squared error, the best candidate is the one with the smallest SSE. Here, Candidate 3 has the lowest SSE of about [tex]$27.29$[/tex], which indicates that it provides the best fit to the data.
Thus, the regression equation that best models the data is
[tex]$$
\boxed{y = 7.57 \sin(-0.55x - 0.68) + 64.40.}
$$[/tex]
This is the regression equation that most accurately represents the given table of average monthly temperatures.
