
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
On December 9, 2021, the "Beijing Compulsory Education Physical Education and Health Assessment Plan" was released. The assessment of compulsory education physical education and health includes two parts: process assessment and on-site examination, with a total score of 70 points. The process assessment is worth 40 points, and the on-site examination is worth 30 points. The assessment plan will be implemented from the date of publication, gradually expanding across different school stages. The on-site examination adopts a classification-limited selection method, dividing the content into four categories, with a total of 22 examination contents including required and elective items. 

In a certain district, 1100 male students and 1000 female students from the ninth grade were randomly selected as samples for a statistical survey. The proportions of male and female students choosing table tennis as an elective item are 10% and 5%, respectively, while the proportions choosing 1-minute jump rope are 40% and 50%, respectively. It is assumed that all students choose each item independently. 

$(Ⅰ)$ From all ninth-grade students in the district, one student is randomly selected to estimate the probability that the student chooses table tennis as an elective item. 

$(Ⅱ)$ Two male students and one female student are randomly selected from all ninth-grade male and female students in the district, respectively, to estimate the probability that exactly two of these three students choose 1-minute jump rope. 

$(Ⅲ)$ Given that the table tennis exam has a full score of 8 points, in a simulated ninth-grade exam in the district, 60 male students who chose table tennis scored 8 points, 40 scored 7.5 points, and the rest scored 7 points; 40 female students who chose table tennis scored 8 points, and the rest scored 7 points. Let the estimated average score of all students who chose table tennis in this simulated exam be $\mu _{1}$, and the estimated average score of male students be $\mu _{2}$. Compare the sizes of $\mu _{1}$ and $\mu _{2}$ without the need for proof.

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