# SD of the predicted value

A variable y is regressed against a single variable x across 28 observations. The value of the slope is 1.89, and the constant is 1.1. The mean value of x is 1.50, and the mean value of y is 3.94. The standard deviation of the x variable is 0.96, and the standard deviation of the y variable is 2.85. The regression sum of squares is 79.07, and the total sum of squares is 195.24. For an x value of 2.0, what is the 95% confidence interval for the y value? A) 0.41 to 9.35. B) 1.01 to 8.75. C) 1.21 to 8.65. D) 1.50 to 8.36. Your answer: A was correct! First the standard error of the estimate must be calculated—it is equal to the square root of the mean squared error, which is equal to the residual sum of squares divided by the number of observations minus 2. The residual sum of squares is equal to the difference between the total sum of squares and the regression sum of squares, which is 195.24 − 79.07 = 116.17. The standard error of the estimate is equal to (116.17 / 26)1/2 = 2.11. ****Where does this formula below come from? Schweser or CFAI.**** The standard deviation of the prediction is equal to the standard error of the estimate multiplied by {1 + 1/n + (X −X )2/[(n − 1)sx2]}1/2 = 2.11 × [1 + 1/28 + (2.0 − 1.5)2/(27 × 0.962)]1/2 = 2.17. The prediction value is 1.1 + (2.0 × 1.89) = 4.88. The t-value for 26 degrees of freedom is 2.056. The endpoints of the interval are 4.88 ± 2.056 × 2.17 = 0.41 and 9.35. --------------------------------------------------------------------------------

I just realized that the formula didn’t format correctly. Sorry about that.

It is the formula for the standard error of forecast (Sf^2)- Reading 11. Sf is used in determining the confidence interval for the dependent variable. Edit: I’m not sure if we need to learn this formula- Schweser says that it is unlikely that we will be asked to calculate this; most probably, it will be given to us.

I hate how they say stuff like that and then ask you questions on how to calculate it. Makes no sense. Thanks. I see it now, page 155 in schweser.

I hear you. Funnily, even I was working out the exact same problem now. What a co-incidence. I wouldn’t worry too much about learning it, really.

The onnly thing you should see about that is that it is significantly wider than the confidence interval for the mean. If you can write down the formula for the prediction interval and the C.I. for the mean, see how they are different, and understand why they should be different, you are doing great.

maratikus came up with a decent way to solve these questions. Assuming you can work out the SEE (Standard error of the estimate), then you just need to remember that the SEP (Standard error of the prediction) is a bit bigger than the SEE. So in this case, if you work out a prediction interval using the SEE (and ignore Joey having an fit in the corner), you get (0.53, 9.23). Given that the true prediction interval will be a bit wider than this, it’s obvious given the choices that the solution is A. No guarantees they won’t try to screw you on the exam, but it’s the method I’ll be using, due to limited room left in my head for complicated formulas.

I wouldn’t have a fit about it - its the way I would do it. Edit: Of course it is slimy and the kharma associated with it will find you holding Enron stock and mezzanine CDO debt.

chrismaths Wrote: ------------------------------------------------------- > maratikus came up with a decent way to solve these > questions. > > Assuming you can work out the SEE (Standard error > of the estimate), then you just need to remember > that the SEP (Standard error of the prediction) is > a bit bigger than the SEE. > > So in this case, if you work out a prediction > interval using the SEE (and ignore Joey having an > fit in the corner), you get (0.53, 9.23). Given > that the true prediction interval will be a bit > wider than this, it’s obvious given the choices > that the solution is A. > > No guarantees they won’t try to screw you on the > exam, but it’s the method I’ll be using, due to > limited room left in my head for complicated > formulas. That is how I did it on this question. It worked well because only A made sense, but I have a feeling we won’t be so lucky on the actual exam. I will definetly look at this shortcut to save some time on the exam. Thanks chrismaths (and maratikus).

Chrismaths I think I am missing something. What am I doing wrong. 3.94 ± 2.056*2.11 = -.039816 - 8.27 I’m a little slow this morning. I understand making it slightly bigger, but I am not getting your .53 - 9.23.

you’re using the mean of y, not the predicted value | X=2 “The constant is 1.1, the value of the slope is 1.89” So plug in an X value of 2, you get 1.1+ 1.89 * 2 = 4.88 So it’s 4.88 ± 2.056*2.11

Ah, Thank you.

We can use SEE only if the number of observations is large so that Sf and SEE converge to the same value. If the same size is small, we can’t use SEE.

chrismaths Wrote: ------------------------------------------------------- > maratikus came up with a decent way to solve these > questions. > > Assuming you can work out the SEE (Standard error > of the estimate), then you just need to remember > that the SEP (Standard error of the prediction) is > a bit bigger than the SEE. > > So in this case, if you work out a prediction > interval using the SEE (and ignore Joey having an > fit in the corner), you get (0.53, 9.23). Given > that the true prediction interval will be a bit > wider than this, it’s obvious given the choices > that the solution is A. > > No guarantees they won’t try to screw you on the > exam, but it’s the method I’ll be using, due to > limited room left in my head for complicated > formulas. Thanks ruhi - this had been pointed out earlier. Shortcut bad karma method is: 1) calculate prediction interval using SEE 2) add a bit. 3) enjoy the extra space left in brain by not memorising the exact formula. I’ve put a deckchair there.

chrismaths Wrote: > Thanks ruhi - this had been pointed out earlier. > > Shortcut bad karma method is: > 1) calculate prediction interval using SEE > 2) add a bit. > 3) enjoy the extra space left in brain by not > memorising the exact formula. I’ve put a deckchair > there. This is what I am going with. If they ask us to calculate this and I miss a point I will be annoyed. Worst case the shortcut should get you to a 50-50 guess.

We can use the time spent in finding roundabout ways to actually just learn the damn formula. Took me only a minute.

I guess…sigh

ruhi22 Wrote: ------------------------------------------------------- > We can use the time spent in finding roundabout > ways to actually just learn the damn formula. Took > me only a minute. But you don’t have a deckchair. I do.

is SEP in the 09 L2?

Yep