
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A sample of 10 families was randomly selected from a residential area, and data was gathered for the monthly income $x_i$ (in thousands of yuan) and monthly savings $y_i$ (in thousands of yuan) of the $i$-th family. The following sums were calculated: $\sum_{i=1}^{10} x_i = 80,$ $\sum_{i=1}^{10} y_i = 20,$ $\sum_{i=1}^{10} x_i y_i = 184,$ $\sum_{i=1}^{10} x_i^2 = 720.$
(I) Find the linear regression equation of monthly savings $y$ with respect to monthly income $x$, $\hat y = \hat b x + \hat a.$
(II) If the monthly income of a family in this residential area is 8000 yuan, predict the monthly savings of this family. Note: In the linear regression equation $\hat y = \hat b x + \hat a$, $\hat b = \frac{\sum_{i=1}^{n} x_i y_i - n\bar{x}\bar{y}}{\sum_{i=1}^{n} x_i^2 - n\bar{x}^2}$, $\hat a = \bar{y} - \hat b\bar{x}$, where $\bar{x}$, $\bar{y}$ are the sample means.

Present the answer in LaTex format: \boxed{Your answer}
user