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Regression line on origin

  1. May 23, 2017 #1
    1. The problem statement, all variables and given/known data

    A random sample of size ##n## from a bivariate distribution is denoted by ##(x_r,y_r), r=1,2,3,...,n##. Show that if the regression line of ##y## on ##x## passes through the origin of its scatter diagram then

    $$\bar y \sum^n_{r=1} x_r^2=\bar x\sum^n_{r=1} x_r y_r$$ where ## (\bar x,\bar y)## is the mean point of the sample.

    I don't really know how to begin. I am aware the line equation is $$b=\frac{y}{x}=\frac{\sum xy-\frac{\sum x\sum y}{n}}{\sum x^2-\frac{(\sum x)^2}{n}}$$

    Not sure what to do next.
     
    Last edited by a moderator: May 24, 2017
  2. jcsd
  3. May 23, 2017 #2

    Ray Vickson

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    (1) Do not write ##\xbar##, write ##\bar{x}##. Right-click on the formula to see its TeX commands.
    Mod note: Fixed the TeX in the original post and above.
    (2) What can you say about the data if the least-squares line has intercept ##a## equal to zero?
     
    Last edited by a moderator: May 24, 2017
  4. May 23, 2017 #3
    Y is proportional to X
     
  5. May 23, 2017 #4

    Ray Vickson

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    That answer is not useful. Take the formula for ##a##, in terms of ##(x_i,y_i)##, and set it to zero. What do you get?
     
  6. May 24, 2017 #5

    SammyS

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    I suppose the that the linear model you are working with is: ##\ \displaystyle y=a+bx\ ##.

    You have the correct expression for finding the linear coefficient, ##\ b\,.\ ##( Leave out the ##\ \displaystyle \frac yx\ ## ).

    It seems to me that you must also consider the expression for ##\ a\,.\ ## Then show that if ##\ a=0\,,\ ## then you obtain the desired result:
    ##\displaystyle \bar y \sum^n_{r=1} x_r^2=\bar x\sum^n_{r=1} x_r y_r ##​
    .
     
    Last edited: May 25, 2017
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