STAT 330 Lecture 32

Reading for Today's Lecture: 12.4, 12.5.

Goals of Today's Lecture:

• Introduce sampling from the bivariate normal population.
• Learn that the regression of Y on X in a bivariate normal population is linear.
• Learn that the conditional distribution of Y given X=x is

  

Today's notes

Correlation Analysis

Our model will be that pairs

are sampled from a bivariate normal population of pairs.

Properties of the bivariate normal density:

• "Ideal Histogram" has shape of (elongated) bell.
• Contours (also called level sets or curves) are (concentric) ellipses.
• If you imagine making a big vertical slice through this bell shaped mountain then the outline has the shape of a normal curve.
• For any constants a and b the variable aX+bY has a normal distribution.
• The bivariate normal density is specified by 5 parameters:
  • Two means: and which are the population means of X and Y.
  • Two standard deviations: and which are the population standard deviations of X and Y.
  • The Correlation Coefficient:

    

Facts about

• (This is called the Cauchy Schwarz inequality by mathematicians.)
• are independent (which works only for the bivariate normal population).
• for some constants a and b with b>0.
• for some constants a and b with b<0.
• The regression of Y on X is linear, meaning

  

• is a unitless (dimensionless) quantity.

Sample Version of this theory:

Definition: the sample correlation coefficient is

(and many other formulae are possible) where

and

Facts about r:

• 
• r=1 if and only if all points lie exactly on an upward sloping line.
• r=-1 if and only if all points lie exactly on an downward sloping line.
• The regression line is

  

• .
• r is a unitless (dimensionless) quantity.

Here are some plots of data sets in which the sample correlation is varied from picture to picture.

Here is a scatterplot in which the correlation is 0 even though X and Y are quite strongly related; the point is that the relation is not a straight line. The best line through the picture is flat.

Here is a simulated data set which illustrates the following points:

• For a bivariate normal population the conditional mean of Y when X is fixed at x is a straight line function of x. We illustrate this as follows:
  • Population of 1000 pairs of Fathers and Sons. Measure X the height of the father and Y the height of the son.
  • , , and .
  • r=0.494.
  • The slope of the regression line is

    

  • The intercept is .
  • For each integer value of x from 62 through 73 I picked out those families in which the father's height was x to within half an inch. So, for example, for x=65 I took all the families where the father was between 64.5 and 65.5 inches tall. Then I averaged the heights of the son's in this group of families, getting 67.36 inches. (There were a total of 62 families in this group.)
  • In the picture below I plotted all the data along with each of the pairs x,average y computed in the last step; these last few points are represented by X's in the plot. I superimposed the regression line.
  • You are supposed to notice that the X's are quite close to the regression line. This is what the regression of Y on X is: a graph of the average value of Y corresponding to a given X.

  
• The conditional distribution of Y given X is normal. The conditional mean is given by the regression line and the conditional variance is (estimated by ). I now picked out the x values within half an inch of 65 inches. Here is a histogram of the heights of the SONS in this set of families:
  

  You are supposed to see that the histogram is reasonably normal. I have superimposed a normal curve whose mean is given by the regression line and whose SD is given by this root mean square error . The curve follows the histogram pretty well.