
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
An experiment was conducted to study the effect of ozone. The experimental design is as follows: $40$ mice were selected, with $20$ randomly assigned to the experimental group and the other $20$ assigned to the control group. The mice in the experimental group were raised in a high-concentration ozone environment, while those in the control group were raised in a normal environment. After a period of time, the increase in body weight (unit: $g$) of each mouse was recorded.
$(1)$ Let $X$ represent the number of specified two mice assigned to the control group. Find the distribution table and mathematical expectation of $X$.
$(2)$ The experimental results are as follows:
The increase in body weight of mice in the control group, sorted from smallest to largest, is:
$15.2, 18.8, 20.2, 21.3, 22.5, 23.2, 25.8, 26.5, 27.5, 30.1$
$32.6, 34.3, 34.8, 35.6, 35.6, 35.8, 36.2, 37.3, 40.5, 43.2$
The increase in body weight of mice in the experimental group, sorted from smallest to largest, is:
$7.8, 9.2, 11.4, 12.4, 13.2, 15.5, 16.5, 18.0, 18.8, 19.2$
$19.8, 20.2, 21.6, 22.8, 23.6, 23.9, 25.1, 28.2, 32.3, 36.5$
$(i)$ Find the median $m$ of the increase in body weight of the $40$ mice. Then, count the number of data points in each sample that are less than $m$ and those that are not less than $m$, and complete the following contingency table:
|              | $< m$ | $\geq m$ |
|--------------|-------|---------|
| Control Group|       |         |
| Experimental Group|   |         |
$(ii)$ Based on the contingency table in $\left(i\right)$, can we be $95\%$ confident that there is a difference in the increase in body weight of mice in a high-concentration ozone environment compared to a normal environment?
Given: $K^{2}=\frac{{n{{(ad-bc)}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$,
| $P(K^{2}\geq k)$ | $0.100$ | $0.050$ | $0.010$ |
|------------------|---------|---------|---------|
| $k$              | $2.706$| $3.841$| $6.635$|

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