
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
"City public transportation" refers to public transportation such as buses and rail transit operating on fixed routes within a city, which is also the main way for people to travel daily. In order to facilitate the travel of residents, a public transportation company in a city has set up a starting point station in a densely populated area. To study the relationship between the interval time $x$ for vehicle departure and the number of passengers waiting $y$, the following data was obtained through investigation:

| Interval Time $(x$ minutes) | $6$ | $8$ | $10$ | $12$ | $14$ |
|-----------------------------|----|----|-----|-----|-----|
| Number of Passengers $(y$ people) | $15$ | $18$ | $20$ | $24$ | $23$ |

$(1)$ Based on the above data, draw a line graph. It is known that a linear regression model can be used to fit the relationship between $y$ and $x$. Please explain using the correlation coefficient.

$(2)$ Establish the regression line equation of $y$ with respect to $x$ and predict the number of passengers waiting when the vehicle departure interval time is $20$ minutes.

Given: For a set of data $(x_{1}, y_{1}), (x_{2}, y_{2}), \ldots, (x_{n}, y_{n})$, the least squares estimates of the slope and intercept of the regression line $\hat{y} = \hat{b}x + \hat{a}$ are $\hat{b} = \frac{\sum_{i=1}^{n}{x_{i}y_{i}} - n\bar{x}\bar{y}}{\sum_{i=1}^{n}{x_{i}^{2}} - n\bar{x}^{2}} = \frac{\sum_{i=1}^{n}{(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i=1}^{n}{(x_{i}-\bar{x})^{2}}}$, $\hat{a} = \bar{y} - \hat{b}\bar{x}$; the correlation coefficient $r = \frac{\sum_{i=1}^{n}{(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i=1}^{n}{(x_{i}-\bar{x})^{2}}\sum_{i=1}^{n}{(y_{i}-\bar{y})^{2}}}}$; $3\sqrt{15} \approx 11.62$.

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