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I have a few related questions regarding Prediction Intervals for outliers (individual data points). The basic formula that I use is: Mean +/- T(-1df)*sd*sqrt(1+1/n), where Mean is the sample mean, T is the Z-score from the Student's T Distribution Table minus 1 degree of freedom, and sd is the sample standard deviation. This formula can be found in many publications, and at Wikipedia's entry on "Prediction Intervals". The formula for Confidence Intervals is the same except for the final part of the formula which is sqrt(1/n) as opposed to sqrt(1+1/n).

**QUESTIONS:**

1. What is the purpose of the last part of these formulas (i.e. sqrt(1/n) for CI and sqrt(1+1/n) for PI? In other words, what do these things seek to adjust? For the CI, it would seem that it's bringing the Standard Error of the Mean into the formula (i.e., sd*sqrt(1/n). But what about the PI--why is it 1+1/n? Is this bringing the Standard Error of the Estimate into the formula. I don't get it. Is it adjusting the sd forward or something?

2. If you wanted to use the Prediction Interval and Confidence Interval formulas and project out five days, would you change the degrees of freedom for the Z-score to -5, and what about the sqrt(1/n) and sqrt(1+1/n). Would you change them to 5/n and 5+5/n? As I state above, I just don't get it.

3. If I have a sample where n=207, and it follows a normal distribution, how many data points should be more than 1.97 sd (.05) below the Mean in a two-tailed test? How many should be more than 2.6 sd (.01) below the Mean?

4. Finally, what is it about the formulas for Confidence Intervals and Prediction Intervals which would cause Sample A to have a narrower Confidence Interval at .05 (in percentage terms) than Sample B, but a wider Prediction Interval at .05 (in percentage terms) than Sample B? The samples are of equal size and both satisfy tests for normality. I don't understand what would cause Sample A to have a narrower confidence interval, but a wider prediction interval. The only thing which is different in the PI formula, as opposed to the CI formula, is the sqrt(1+1/n) instead of sqrt(1/n).

Thank you.