Review the table showing average monthly temperatures in degrees Fahrenheit for several months, where [tex]$x=1$[/tex] represents January.

\[
\begin{tabular}{|c|c|}
\hline
Month & \begin{tabular}{c}
Average Monthly Temperature \\
( [tex]${ }^{\circ}F$[/tex] )
\end{tabular} \\
\hline
1 & 58 \\
\hline
3 & 60 \\
\hline
4 & 64 \\
\hline
6 & 68 \\
\hline
7 & 70 \\
\hline
8 & 73 \\
\hline
9 & 71 \\
\hline
12 & 55 \\
\hline
\end{tabular}
\]

Which regression equation best models the data?

A. [tex]y=7.57 \sin (0.55 x + 0.68) + 64.40[/tex]

B. [tex]y=7.57 \sin (0.68 x + 0.55) + 64.40[/tex]

C. [tex]y=7.57 \sin (-0.55 x - 0.68) + 64.40[/tex]

D. [tex]y=7.57 \sin (-0.68 x - 0.55) + 64.40[/tex]

To determine which regression equation best models the given data, we need to evaluate each provided equation and find out which minimizes the difference between the observed and predicted temperatures. This approach involves calculating the sum of squared residuals (the differences between the actual temperatures and the temperatures predicted by the model) for each equation.

Here are the candidate regression equations:

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]

### Step-by-Step Solution:

1. Data Points:
- (1, 58)
- (3, 60)
- (4, 64)
- (6, 68)
- (7, 70)
- (8, 73)
- (9, 71)
- (12, 55)

2. Calculate the sum of squared residuals for each model:

- For [tex]$ y = 7.57 \sin(0.55x + 0.68) + 64.40 $[/tex]:
- Sum of squared residuals: 1161.007506753333

- For [tex]$ y = 7.57 \sin(0.68x + 0.55) + 64.40 $[/tex]:
- Sum of squared residuals: 854.9806516230036

- For [tex]$ y = 7.57 \sin(-0.55x - 0.68) + 64.40 $[/tex]:
- Sum of squared residuals: 27.29271110195468

- For [tex]$ y = 7.57 \sin(-0.68x - 0.55) + 64.40 $[/tex]:
- Sum of squared residuals: 168.8183544044771

3. Identify the model with the smallest sum of squared residuals:

The model [tex]$ y = 7.57 \sin(-0.55x - 0.68) + 64.40 $[/tex] has the smallest sum of squared residuals (27.29271110195468), indicating it best fits the data.

Therefore, the regression equation that best models the data is:
[tex]\[ y = 7.57 \sin(-0.55x - 0.68) + 64.40 \][/tex]