Edited: In relation with our research, the respondents were deaf employees but due to the difficulty to have a large sample size, we're hoping to target that the respondents will be n=40 to reasonably say that it is a good estimate of statistical inference of the data for prediction using a simple linear regression analysis. And so, how many should be the participants to say that it is a good sample to have a good prediction? -like to have a good correlation, from my knowledge, (is it true? that n= 30 is the smallest good sample size) then, what about for a simple linear regression?
closed as unclear what you're asking by Michael Chernick, kjetil b halvorsen, mdewey, Peter Flom - Reinstate Monica♦ Jul 28 '18 at 14:22
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There are two different issues at play here. This is what we often call "sample size planning." It depends what your goal is here. Broadly-speaking, we could be doing a linear regression to get predictions for the future, or we could be trying to make some type of statistical inference.
I'm afraid that these issues are too broad of a question—there have been books written on this area. However, a few tips that will help you get the answer you want:
If you are interested in prediction, I would look into books about "statistical learning" and "machine learning" that cover simple linear regression. Those books are more focused on accuracy of predictions, and many discuss necessary sample sizes to get the accuracy you wish.
If you are interested in statistical inference, I would look into "power analysis." This area is focused on: "If we have this sample size, how often would we observe a significant effect, given that that significant effect exists with some given strength?" There are a lot of books and articles in the social sciences that cover power analysis, especially when it comes to linear regression.
Note that, unfortunately, the answer will not be straightforward. There is not going to be a threshold everyone agrees on. It depends on your goals, the effect sizes you are researching, etc.