Correlation Suppose that the average lifespan for people who smoke is:
We can calculate the least squares regression line: y = 73  1.3x We define the first residual to be the difference between the first lifespan and the first estimated lifespan: 72  (73  1.3(1)) = 0.3 the second residual as: 70  (73  1.3(2)) = 0.4 the third as: 69  (73  1.3(3)) = 0.1 and the fourth as 68  (73  1.3(5)) = 1.5 in general we have the residual is
Coefficient of determination: r^{2} We define the coefficient of determination as an indication of how linear the data is. r^{2} has the following properties:
Properties of the Coefficient of Determination
Construction To compute r^{2}, do the following:
If we multiply r^{2} by 100%, we arrive at the percent of the observed variation attributable to the linear relationship.
Correlation: r If we want to determine not just if they are linearly related, but also want to know whether there is a positive relationship or a negative relationship (b> 0 or b<0) and want the calculation unitless, we compute Pearson's correlation coefficient r
We have r^{2} = r^{2} that is the square of the correlation coefficient is equal to the coefficient of determination.
We say that the correlation is
For example there may
be a strong correlation between grayness in hair and wrinkles, but having
gray hair does not cause one to have wrinkles.
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