So I've constructed a confidence interval for my regression line. However, because I have 2500 data points it is a very, very narrow interval (I can barely see it next to the regression line when I make it a 99% interval). I've also been reading similar questions but I just can't get a basic understanding of what this confidence interval means.
Here are my thoughts..
I think it is saying that if I draw another 2500 sample that I can be 99% sure the new estimated regression line will lie within this C.I? Or, in other words, I could draw 100 more samples, fit them and expect only 1 out of the 100 estimated regression lines would violate these new CI. I notice it widens at each end of the graph which would allow a new line to pivot properly if a slightly different coefficient was estimated so that sort of makes sense.
Please let me know what you think I would really appreciate it!
Best Answer
You are partially correct about the interpretation of confidence interval. It tells about the population parameter (as opposed to sample statistic) which in your case is regression coefficient of the entire population. A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter such as mean and regression coefficient. Because of their random nature, it is unlikely that two samples from a given population will yield identical confidence intervals. But if you repeated your sample many times, a certain percentage of the resulting confidence intervals would contain the unknown population parameter. The percentage of these confidence intervals that contain the parameter is the confidence level of the interval.
A $99\%$ confidence interval indicates that $99$ out of $100$ from the same population will confidence intervals that contain the population parameter. To make the idea clearer consider this: As the sample size increases, the sampling error decreases and the intervals become narrower. If you could increase the sample size to equal the population, there would be no sampling error. In this case, the confidence interval would have a width of zero and be equal to the true population parameter.
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