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Below, I'm showing a hypothetical simple linear regression case. There are $5000$ datapoints in each of the 4 sub-populations stacked over the top of each predictor value. Each subpopulation is normally distributed. And all subpopulations have the same $\sigma^2$. However, the subpopulations' $\mu$ are different leading to an upward-sloping regression line.

Question

When I compute the $99.999999999\%$ confidence interval for the regression line, nothing basically shows (i.e., the line is super precise due to huge number of datapoints).

However, When I compute the $99.999999999\%$ prediction interval for predicting a future observation (the light green surrounding the 4 subpopulations as well as the regression line), the prediction interval becomes hugely wide.

Why when datapoint are so huge (about the size of a population), setting the confidence interval around the regression line to about $1$, doesn't make the such confidence interval to even show BUT

setting the prediction interval for observations to about $1$, is still so huge? (Is there really any other far-away observation than the ones currently existing in each subpopulation)

enter image description here

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  • $\begingroup$ It's difficult to determine what you are asking. You seem to want to set limits that are likely to contain nearly every point in a sample of size $10^{11}$. Are your "subpopulations" really that large? $\endgroup$ – whuber May 19 '17 at 17:58
  • $\begingroup$ @whuber, so where does 10^11 come from? $\endgroup$ – rnorouzian May 19 '17 at 18:01
  • $\begingroup$ @whuber, are you suggesting that a "subpopulation" should be extremely large to contain just about all possible observations? $\endgroup$ – rnorouzian May 19 '17 at 18:05
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    $\begingroup$ $10^{11} = 1 / (1 - 99.999999999/100)$ is from your interval. It's looking like you might benefit from reviewing our threads on confidence intervals and prediction intervals: I recommend you search the site. $\endgroup$ – whuber May 19 '17 at 18:16
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The 99.999999999% confidence bands defines the area that, with repeated experiments, contain the true regression line 99.999999999% of the time. With lots of data, you've defined the regression line very precisely, so that band is super narrow.

The 99.999999999% prediction bands define the area that you expect to contain 99.999999999% of future data points you collect. To be so sure (so many 9s), those bands are really wide. The sample size doesn't affect the prediction bands so much.

Of course this is based on all the assumptions of linear regression.

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  • $\begingroup$ Harvey, @whuber mentioned that, I need the size of my subpopulations each to be $10^{11}$ (i.e., $1 / (1 - 99.999999999/100)$ ), How this simple formula comes about? Is this formula only true in the case of normal distributions? $\endgroup$ – rnorouzian May 19 '17 at 20:48
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    $\begingroup$ @huber is calculating the number of data points you'd need to collect to expect one to be outside the prediction bands. This points out how pointless it is to compute prediction bands with so many nines. $\endgroup$ – Harvey Motulsky May 19 '17 at 22:23

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