
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 random sample of $10$ households was taken from a residential area, and the monthly income $x_{i}$ (in thousand yuan) and monthly savings $y_{i}$ (in thousand yuan) of the $i$-th household were obtained. The following results were calculated: $\sum_{i=1}^{10}{x}_{i}=80, \sum_{i=1}^{10}{y}_{i}=20, \sum_{i=1}^{10}{x}_{i}{y}_{i}=184, \sum_{i=1}^{10}{{x}_{i}}^{2}=720$

(1) Find the linear regression equation $y=bx+a$ of monthly savings $y$ on monthly income $x$.

(2) Determine whether the variables $x$ and $y$ are positively or negatively correlated.

(3) Predict the monthly savings for a household in this residential area with a monthly income of $7$ thousand yuan.

Note: In the linear regression equation $y=bx+a$, $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}},a= \bar{y}-b \bar{x} $, where $\bar{x}, \bar{y}$ are the sample means.

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