
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
Vehicle exhaust contains pollutants, and the concentration of pollutants in the exhaust emitted by vehicles will increase after several years of use. Therefore, based on the use and safety technology of motor vehicles and emission inspection conditions, the country implements mandatory scrapping for motor vehicles that meet the scrapping standards. In order to understand the public's understanding of the mandatory scrapping standards for motor vehicles, an environmental organization randomly surveyed 100 people, and the data obtained is shown in the following contingency table:

|          | Unfamiliar | Familiar | Total |
|----------|------------|----------|-------|
| Female   | 20         | 30       | 50    |
| Male     | 10         | 40       | 50    |
| Total    | 30         | 70       | 100   |

$(1)$ Is there a $95\%$ confidence that "understanding of the mandatory scrapping standards for motor vehicles is related to gender"?

$(2)$ The environmental organization obtained data on the number of years $t$ of use of a certain model of car and the percentage $y\%$ of $CO$ concentration in the emitted exhaust as shown in the table below:

| $t$ | 2   | 4   | 6   | 8   | 10  |
|-----|-----|-----|-----|-----|-----|
| $y$ | 0.3 | 0.3 | 0.5 | 0.7 | 0.8 |

If the number of years of use of this model of car does not exceed 12 years, it can be approximately considered that $y$ is linearly related to $t$. Determine the linear regression equation for $y$ with respect to $t$.

Reference formula: ${K}^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$, where $n=a+b+c+d$.

| $P(K^{2}\geqslant k_{0})$ | 0.10 | 0.05 | 0.025 | 0.010 | 0.005 | 0.001 |
|---------------------------|------|------|-------|-------|-------|-------|
| $k_{0}$                   | 2.706| 3.841| 5.024 | 6.635 | 7.879 | 10.828|

In the linear regression equation $\hat{y}=\hat{b}t+\hat{a}$, $\hat{b}=\frac{\sum_{i=1}^{n}{{t}_{i}}{y}_{i}-n\overline{t}\overline{y}}{\sum_{i=1}^{n}{{t}_{i}^{2}}-n\overline{{t}}^{2}},\hat{a}=\overline{y}-\hat{b}\overline{t}$.

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