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[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

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In the autumn of 2023, Mycoplasma pneumoniae pneumonia was prevalent in various parts of the country. The main infected groups of this disease are teenagers and the elderly. The infectious disease department of a certain city hospital randomly surveyed $200$ elderly people aged $70$ and above who sought medical treatment in various hospitals in the city during a certain period of time, and organized the survey results as shown in the table below:

|                      | With Chronic Disease | Without Chronic Disease | Total |
|----------------------|----------------------|-------------------------|-------|
| Not infected         | $60$                 | $80$                    | $140$ |
| Infected             | $40$                 | $20$                    | $60$  |
| Total                | $100$                | $100$                   | $200$ |

$(1)$ Can we be $99.5\%$ confident that the infection of Mycoplasma pneumoniae pneumonia in elderly people aged $70$ and above is related to having chronic diseases?<br/>$(2)$ Now, $6$ elderly people out of the $60$ infected with Mycoplasma pneumoniae pneumonia are sampled in a stratified manner. Then, $4$ people are randomly selected from the $6$ people as subjects for medical research and treatment. Based on past experience, the research and treatment cost per person with chronic diseases is $20,000$ yuan, and for those without chronic diseases, it is $10,000$ yuan. Let $\xi$ denote the total cost of research and treatment incurred by these $4$ individuals. Find the probability distribution and mathematical expectation of $\xi$.<br/>Reference table:<br/><table><tbody><tr><td>$P(K^{2}\geqslant k)$</td><td>$0.15$</td><td>$0.10$</td><td>$0.05$</td><td>$0.025$</td><td>$0.010$</td><td>$0.005$</td><td>$0.001$</td></tr><tr><td>$k$</td><td>$2.072$</td><td>$2.706$</td><td>$3.841$</td><td>$5.024$</td><td>$6.635$</td><td>$7.879$</td><td>$10.828$</td></tr></tbody></table>Reference formula: ${K}^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$ (where $n=a+b+c+d$)

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