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  Theory of Regression

Regression:

The basic idea behind regression is that whenever there is significant linear correlation, we can employ a line to approximate the value of dependent variable for some values of the independent variable.

The regression equation must be used only:

a) Whenever there is significant linear correlation. That is, whenever you refuse the null hypothesis which rho=0 in a correlation hypothesis test.

b) The value of independent variable being employed in the estimation is close to the original values. That is, you must not use a regression equation obtained by using x's between 10 to 20 to estimate y whenever x is 200.

c) The regression equation must not be employed with different populations. That is, when x is the height of a male and y is the weight of a male, then you shouldn't employ the regression equation to find out the weight of a female.

d) Regression equation shouldn't be employed to forecast the values not from that time-frame. When data is from 1960's, it probably is not valid in the year 1990's.

Supposing that you have decided that you can have a regression equation because there is significant linear correlation among the two variables, the equation becomes: y' = ax + b or y' = a + bx. The Bluman text employs the second formula, though, more people are known with the notion of y = mx + b.

a is the slope of regression line:

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b is the y-intercept of regression line:

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The regression line is sometimes termed as the ‘line of best fit’ or the ‘best fit line’.

As it ‘best fits’ the data, it makes the sense that line passes via the means.

Regression equation is a line with slope passing via the point:

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The other way to write the equation would be:

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Apply just a little algebra, and we encompass the formulas for a and b which we would use (when we were stranded on the desert island devoid of TI-82).

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It as well turns out that the slope of regression line can be written as a = r (sy/sx). As the standard deviations cannot be negative, the sign of slope is found out by the sign of correlation coefficient. This agrees with the statement made prior that the slope of regression line will encompass similar slope as the correlation coefficient.

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