A Standard Error Question

Question

INFO GIVEN: A paper summarized a regression of y = percentage of pups in a capture on x = percentage of CPV prevalence among adults and pups. The equation of the least-squares line, based on n = 10 observations, was y = 62.804 - 0.53568x, with r2 = 0.55.

QUESTION I NEED TO ANSWER: Suppose that SSTo = 2520 (this value was not given in the paper). What is the value of se? (Give the answer to two decimal places.)

Thank you in advance!!

Answer 1

First, you already know R² and they've given you the total sum of squares SSTO, which allows you to calculate the SSR, the sum of the squares for the regression.   

 

R² = SSR / SSTO = 0.55

SSR = 2520 * 0.55 = 1386

 

From there, you can get the the SSR's "inverse", the sum of the squares of the residuals (errors), called SSE. 

 

SSE = SSTO - SSR = 2520 - 1386 = 1134

 

Now, since that is the sum of the squares of the errors, we can now calculate the mean of the square of the errors, the MSE.  The only tricky part of this is remembering to subtract 2 from the number of observations before you divide. You can think about that "minus 2" as being due to the fact that the two ends of the plot line don't count as differences.  It's not exactly accurate as to the why, but it's a good memory trick.    

 

MSE = (SSE) / (N-2) = 1134 / 8 = 141.75

 

Since that's the Mean of the Squared errors, we need to square root it to get the standard error.  

 

SE = SQRT(MSE) = SQRT(141.75) = just under 12 - you do the math to 2 digits.

 

 

I'm assuming that these are whole percents that we are reporting, since X and Y are described as "percentage" and the fit line goes through (0,62.804);

 

Now, doing our sanity checking on our answers, this says that captures of (whatever kind of animal this is) without pups had between 51 and 76 percent CPV prevalence (62.8 +/- 12) and captures of all pups had between 0 and 22 percent CPV prevalence (62.8 - 53.6  +/- 12) .  

 

If this matches the expected fit, then we're golden.