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The way I learned about Central Limit Theorem in school is illustrated in the following example:

  • Suppose you have a population of 100,000 basketball players. You are interested in knowing the average height of all these basketball players - but you only have the time and money to measure 5,000 of these players. If you measured a sample of 5,000 randomly chosen player and took the average height, and if your friend also a sample of 5,000 randomly chosen players and took the average height of his sample .... if enough people were to do this, the average of all these samples would have a Normal Distribution and this average (in theory) would be the same as the true average height of all basketball players. If I understand correctly, these last two properties are the result of the Central Limit Theorem (CLT).

I am interested in understanding if the Central Limit Theorem can also be used to make inferences about coefficients from a Regression Model. For example - consider the following table:

enter image description here

I remember hearing somewhere that the p-value for each regression follows a T-Distribution. We can use the T-Distribution to estimate the significance of this coefficient (both in isolation as well as along with the other coefficients in the model). This leads me to my question:

Suppose there is a dataset (e.g. we want to predict "salary" based on "height", "age" and "weight") that contains the entire population of some country. Suppose I take a random sample from this population dataset and fit a specific type of regression model to this data (e.g. y = b_0 + b_height + b_age + b_weight ... linear regression, no interaction terms, etc.), suppose my friend takes a random sample from this same population dataset and also fits the same type of regression model to this data. Imagine many people repeat this process and fit regression models to their random samples.

  • Does Central Limit Theorem argue that if all of us were to make a histogram for each regression coefficient (e.g. b_height), would these histograms have a Normal Distribution?

  • Does Central Limit Theorem argue that provided we had sufficient reasons to believe that all of our samples were "sufficiently random" (not sure how we would quantify this randomness) - suppose if we selected "b_weight" from any of the models : Would each "b_weight" represent the "true" impact of weight on salary for all individuals within the population according to our model?

Thanks!

References:

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    $\begingroup$ You can find an at least partial answer at stats.stackexchange.com/questions/521215/… $\endgroup$
    – kjetil b halvorsen ♦
    Jul 6, 2022 at 23:14
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    $\begingroup$ Your characterization of the CLT is incorrect, so reviewing the CLT would be a good place to start. And if you would like to apply the CLT to regression, you have a fundamental issue to address at the outset: what exactly would be your sequence of random variables? That is, in what way are you going to model the collection of more and more data? $\endgroup$
    – whuber ♦
    Jul 6, 2022 at 23:17
  • $\begingroup$ @ whuber: thank you so much for your reply! Can you please help me understand where my understanding of CLT is incorrect? Thank you so much! $\endgroup$
    – stats_noob
    Jul 7, 2022 at 0:55

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