Scientists want to find the relation between two variables $x$ and $y$ by line
$$y=ax+b.$$
After some experiments, scientist $S_i$ find the best parameters are $(a_i, b_i)$. It is
sure that different scientists find different parameters.
The question is what can we say about $(a_i, b_i)$ for a large number of scientists?
(a friend of mine asked this to me, I am not doing this kind of maths:( I suppose this is related to statistics or probability.
What you propose $$y=\hat{a}x+\hat{b}$$ is a simple linear regression, where $\hat{b}$ and $\hat{a}$ are parameters to be estimated from the data. This is indeed what many empirical scientists do (e.g., psychologists doing t-tests with experimental data). Suppose that, whatever they experiment on, the true relationship ("data generating process") really takes the form $$y=ax+b$$ for some $a, b\in\mathbb{R}$. Then the simple linear regression estimates are unbiased under some regularity assumptions. An unbiased estimate means that $$E[\hat{a}]=a,~E[\hat{b}]=b.$$ Thus, under these assumptions, if all scientists sample their data randomly, then averaging all estimates gives $$\bar{a}=n^{-1}\sum_{i=1}^n a_i\to a \text{ as } n\to\infty,$$ with probability 1, by the strong law of large numbers. In other words, if you have sufficiently many independent "studies", then the average should be close to the true values. Also, each scientist is "right" on average. Moreover, by the central limit theorem, $\bar{a},~\bar{b}$ are normally distributed with mean $a,b$ for sufficiently large $n$. Thus, for large $n$, it is very unlikely that the average estimate is far from the true value.
This is the ideal theory world, of course, and in reality there are some obstacles. The biggest problem is that the world is not simple and linear so that $$y=ax+b$$ holds. And if it doesn't, then even infinitely many independent studies will not help to find the true relationship between $y$ and $x$, because scientists are simple fitting the data into the wrong functional form. There are nonparametric techniques to tackle problems where the functional form $f$ in $y=f(x)$ is not known, but this is just starting (and needs much more data than parametric techniques).
Another problem is scarcety of data. For example, in economics there is only one GDP per year, and US data doesn't even reach back to the 19th century. In other areas, like physics, this may not be a problem, because data can be created in the lab. The problem there is more likely manipulation of data.
