Math 151 , Spring '08, Mon. Day 16, March 3,Hit reload.. .After class. corrections to A

Formula yhat = a + b x.    Govsal = a + b avgpay   Govsal = 28,569.69 + 2.709*avgpay
     yhat =  28,569.69 + 2.709* x        
         To predict or estimate a y-value for a given x-value, plug the x value into the formula and calculate.
                To do it graphically, use the Up-and-Over method (Fig. 5.1, p.116):
                    Find the x, go straight up to the line, then go over to the y-axis; that y-value is the predicted y.
         Calculating:  Montana (17,895, 55,502)   Govsal = 28,569.69 + 2.709*avgpay
           Predicted Govsal = 28,569.69 + 2.709*17,895 = 28,569.69 48,477.56 = 77,047.25 (higher than actual)

(Graphing a straight line:  pick an x-value at one end of the useful range.  Plug in to the formula and calculate the corresponding y.  Graph the (x,y) pair.  Repeat with an x value at the other end of the range.  Connect the 2 dots with a line (see pretest).  Insurance:  Pick a third x and calculate the y.  This point must also lie on the line, if you did it right.)

 a is y-intercept. b  is slope:  If x increases one unit, yhat increases b units.  
  If you know that yhat increases 12 units for every one that x increases, you know that the slope of the line b = 12. 
            Governor's salaries increase (on the average across the states)  $2.71 for every increase of  $1 of average pay.
     This is a summary  of the linear relationship, in the same way that the mean of a distribution is one summary of the distribution.  Particular states won't match this exactly.

 (In a straight-line relationship, the amount that y increases for one unit increase in x is the same no matter what value of x you start with)  RegressionSlope.xls or in ClassMaterial\Math151-BPS4e \RegressionDemos Excel BPS4e

Income depends on height?!
    What is "$789", and what kind of analysis did they do?  (HW)

We all get the same line from a batch of data because we use the "least-squares best fit" criterion (p. 119): we'll investigate this more closely later.

Facts:  1, 2 lite, 3 first.  Then 4.   Then 2 &Formulas p. 120, from 2&3.  

Facts (Moore pp. 123-125)

1. Which is explanatory, which is response, is crucial for regression!  The Regression line is trying to predict the "average y" for a given x (with the added requirement that it is a straight line).  See "residual"(deviation) lines for govsal on avgpay.
   Unless the data lies perfectly on a straight line, the line for predicting weight from height -- "regressing weight on height" --(for example) will NOT be the same line as that for predicting height from weight--"regressing height on weight".  (In-class demonstration,soon, on overhead projector.) (Example 5.3, Fig. 5.4 pp.123-4 is about this. )
    
2. Lite:  The correlation coefficient r and the slope b of the regression line have the same sign!  + or - .
      Negative/positive:  trend=slope ~association~correlation
   Heavy: A change of one standard deviation in x corresponds to a change of r standard deviations in y, along the regression line.  We'll return to this.
   


3. The regression line goes through the point given by the two means, (xbar, ybar). 
   Applet http://www.whfreeman.com/bps4e
   We'll return to this.
   


4. r² ("Coefficient of Determination") = fraction of the variation in y-values explained/predicted by knowing x and using the least squares regression line.  SPSS writes "R Square", or "R Sq Linear".  (Exactly what that means mathematically is hard.  Just get used to it as a measurement.)
   Closer to 0, more scatter around the line. Closer to 1, tighter clustering around the line. R-Squared (or RSquared.xls: ClassMaterial\Math151-BPS4e\RegressionDemosExcelBPS4e)(Optional:  Further explanation of r_²)

r² is the square of the correlation coefficient r!  (-, + Sign gets lost.)  
If r = .7, about half (.49) of the variation  in the y's is explained by using the regression line relationship to predict y from x. (If weight and height have a correlation of .7, then half of the variability in weight can be explained by knowing height. Or vice versa.)
NOTE:  The standard deviation doesn't say anything about the distance of any individual point from the mean; it's only about a kind of "average" variability. 
R² doesn't say anything about the line and any particular (x,y) pair --just about a kind of "average" goodness of the explanatory power of the line for the data.