Recent questions in Population Data

Recent questions in Population Data
Population Data Answered
Aron Heath 2022-11-17

Is a t distribution for a certain degree of freedom equivalent to the sample mean distribution for the corresponding sample size?
This may seem like a weird question, but hear me out. I'm essentially struggling to see the connection between a t-value from a t-table and a t-value that is calculated.
The following formula is used to calculate the value of a t-score:
t = X ¯ μ S n
It requires a sample mean, a hypothesized population mean, and the standard deviation of the distribution of sample means (standard error).
According to the Central Limit Theorem, the distribution of sample means of a population is approximately normal and the sample distribution mean is equivalent to the population mean.
So the t-score formula is essentially calculating the magnitude of difference between the sample mean in question and the hypothesized population mean, relative to the variation in the sample data. Or in other words, how many standard errors the difference between sample mean and population mean comprise of. For example: If t was calculated to be 2, then the sample mean in question would be 2 standard errors away from the mean of the sample distribution.
1.) Phew, ok. So question 1: Let's just say a t-score of 1 was calculated for a sample mean and since a distribution of sample means is normal according to the CLT, does that mean that the sample mean in question is part of the 68% (because of the 68 95 rule)of all sample means that are within 1 standard error of the sample mean distribution?
2.) Let's say we have a distribution of sample means of sample size 15. Is this distribution equivalent for a t-distribution of degrees of freedom 14? Or more importantly: Is the t-value from a t-table for 14 degrees of freedom and 95 confidence EQUIVALENT to a calculated t-value using a sample mean that is 2 standard errors away from the mean of a distribution of sample means with sample size 15?

Population Data Answered
vidamuhae 2022-11-14

COVID19 data statistical adjustment for SIR model and estimation
All of us are coping with the current COVID19 crisis. I hope that all of you stay safe and that this situation will end as soon as possible.
For this sad situation and for my unstoppable curiosity, I've started to read something about the SIR model. The variables of such model are s (the fraction of people susceptible to infection), y (the fraction of infected people) and r (the fraction of recovered people + the sad statistics of deaths). The model reads as:
{ s ˙ = β s y y ˙ = β s y γ y r ˙ = γ y ,
where β and γ are positive parameters. One strong hypothesis of this model is that the population size is constant over time (deaths are assumed to be recovered, births are neglected since, hopefully, they will be the part of the population which for sure will be protected from the disease). The initial conditions are set such that s ( 0 ) + y ( 0 ) + r ( 0 ) = 1 and s ( 0 ) 0, y ( 0 ) 0 and r ( 0 ) 0. Under this assumption, it can be proven that s ( t ) + y ( t ) + r ( t ) = 1   t > 0.
The news often talk about the coefficient:
R 0 = β γ ,
which rules the behavior of the system (for R 0 < 1 the disease will be wiped out, for R 0 > 1 it will spread out).
The same news also talk about the estimation of such parameter. Well, given the time series of s, y and r, it is rather easy to estimate the parameters β and γ, and hence R 0 . My main concern is about the time series. For each country we know the daily count of infected people (let's say Y(t)), of recovered (or dead) people (let's say R(t)).
Anyway, there are several infected people which are not recorded (let's say Y′(t)), and many of them get recovered without knowing that they have been infected (let's say R′(t))! Moreover, day after day, the number of tests on people is increasing.
If we indicate with N the (constant) size of population, we get that:
y ( t ) = Y ( t ) + Y ( t ) N , r ( t ) = R ( t ) + R ( t ) N   and   s ( t ) = 1 y ( t ) r ( t ) .
Here is the question(s). How can we perform the estimation of β and γ if we don't know the unobserved variables Y′(t) and R′(t)? How do the experts of the field estimate β and γ even though the available data are not complete? Do they use some data adjustment?