
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]


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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. The results of the experiment are as follows:

The increase in body weight of the mice in the control group, sorted from smallest to largest, is as follows:

| 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 the mice in the experimental group, sorted from smallest to largest, is as follows:

| 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 |

$(1)$ Calculate the sample mean of the experimental group.  
$(2)$  
$(i)$ 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 greater than or equal to $m$, and complete the following contingency table:

|         | $< m$ | $\geq m$ |
|---------|-------|---------|
| Control |       |         |
| Experimental |       |         |

$(ii)$ Based on the contingency table in $(i)$, 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: $K^{2}=\frac{{n{{(ad-bc)}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$,  
Contingency Table:

| $P(K^{2}\geq k)$ | $0.100$ | $0.050$ | $0.010$ |
|------------------|---------|---------|---------|
| $k$              | $2.706$ | $3.841$ | $6.635$ |

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